My friend David Miller is looking for instructors to help out with the Newton Girls who Code club. Here's an announcement, please connect with him if you're interested.

They visited Harvard last week -- David gave me permission to post his description of the visit. It seemed like a well-organized event -- thanks to the Harvard Women in Computer Science group for putting it together.

----

Last Friday, the Newton Girls Who Code club was welcomed by the Harvard Women in Computer Science at Harvard's Maxwell-Dworkin Laboratory. The students learned about the joint Harvard-MIT Robot Soccer team
from the mechanics tech lead, Harvard junior Kate Donahue. She showed
us last years fleet of robots (named after Greek and Roman gods), and
described their work on preparing this years fleet (to be named after
Harry Potter characters). Kate emphasized the interplay between computer
vision, artificial intelligence, mechanical engineering, and
distributed systems. Many of the robot parts are 3D printed -- a
technology that the Newton GWC students will become more familiar with
this fall as we integrate the Newton Library's 3D printer into the GWC
activities.

After
the robots demonstration, the students took part in a Q+A
discussion with Harvard WiCS undergrads Hana Kim, Ann Hwang, and Adesola
Sanusi. Our students asked great questions about our hosts' motivation
and history with coding, the mechanics of being a college CS major, the
role of gender in the CS community, the connections between computer
science and other fields, and our hosts' vision for the future of
computing. The WiCS undergrads are excellent role models and were
enormously encouraging. They pronounced our students more than ready to
take Harvard's most popular course, Introduction to Computer Science, and recommended they try out the free, online, EdX version today. It was an exhilarating afternoon!

## Saturday, April 25, 2015

## Monday, April 20, 2015

### My (Positive) Apple Retail Repair Experience

I switched to Apple machines several years ago, and have been very happy with them generally.

One convenience is there is an Apple store in the mall about 10 minutes from my home, so when things go wrong, I know where to go. Thursday night a screen-related hardware glitch developed (which, to be clear, in this case was not what I would call a product defect; let's say it was a me defect). I ran over and they said it would probably need to be taken in for a repair.

Not being ready to hand over machine just yet, I went home to pull off some data, and then booked an appointment. The downside to the Apple store is you really need to book an appointment online, even for (or especially for) things involving repair. My walk-in Thursday I didn't even get up to the Genius Bar; one of the people up front just suggested what they thought was going on. And my Apple store, at least, is painfully busy, all the time. I got an appointment for Saturday afternoon, nothing available Friday. I went in Friday morning before heading to work on the off chance I could get it in then; they said the walk-in waiting time was 90 minutes to 2 hours. I should have expected; the walk-in waiting time always seems to be measured in hours.

That's the downside. The positive is, when you have an appointment, they are great. I had no wait when I went in Saturday, and in the past, if they're running behind, they've let me know if I have to wait for my appointment and estimated for how long. They are courteous and professional. They will let you know what they are doing as they test your machine, what needs to be done, and provide estimates for how long it will take. Their setup really engenders trust. I also have found their repair times reasonable and understandable. My machine is back to me and all is well Monday afternoon. (Thank you, Apple Store.)

I haven't had to deal with them often over the last decade -- I'd say a reasonable number of times considering the number of Apple machines I've had and in family use. Overall this experience is typical. There's anxiety in going in for any sort of repair process, but they did an excellent job.

Finally, I imagine this is not useful advice to people reading this blog, but do back up your data regularly and/or automatically. One of the first questions they always ask is whether you have a backup, and it's always a good feeling to know that if worst comes to worst you just put your state on a new machine. Someone else at the Genius Bar while I was there had their computer put back in working order but their data was lost; they weren't the sort of user that it was a real problem, but the occasional reminder that backups are important is always helpful.

One convenience is there is an Apple store in the mall about 10 minutes from my home, so when things go wrong, I know where to go. Thursday night a screen-related hardware glitch developed (which, to be clear, in this case was not what I would call a product defect; let's say it was a me defect). I ran over and they said it would probably need to be taken in for a repair.

Not being ready to hand over machine just yet, I went home to pull off some data, and then booked an appointment. The downside to the Apple store is you really need to book an appointment online, even for (or especially for) things involving repair. My walk-in Thursday I didn't even get up to the Genius Bar; one of the people up front just suggested what they thought was going on. And my Apple store, at least, is painfully busy, all the time. I got an appointment for Saturday afternoon, nothing available Friday. I went in Friday morning before heading to work on the off chance I could get it in then; they said the walk-in waiting time was 90 minutes to 2 hours. I should have expected; the walk-in waiting time always seems to be measured in hours.

That's the downside. The positive is, when you have an appointment, they are great. I had no wait when I went in Saturday, and in the past, if they're running behind, they've let me know if I have to wait for my appointment and estimated for how long. They are courteous and professional. They will let you know what they are doing as they test your machine, what needs to be done, and provide estimates for how long it will take. Their setup really engenders trust. I also have found their repair times reasonable and understandable. My machine is back to me and all is well Monday afternoon. (Thank you, Apple Store.)

I haven't had to deal with them often over the last decade -- I'd say a reasonable number of times considering the number of Apple machines I've had and in family use. Overall this experience is typical. There's anxiety in going in for any sort of repair process, but they did an excellent job.

Finally, I imagine this is not useful advice to people reading this blog, but do back up your data regularly and/or automatically. One of the first questions they always ask is whether you have a backup, and it's always a good feeling to know that if worst comes to worst you just put your state on a new machine. Someone else at the Genius Bar while I was there had their computer put back in working order but their data was lost; they weren't the sort of user that it was a real problem, but the occasional reminder that backups are important is always helpful.

## Friday, April 03, 2015

### Women in Math (and Computer Science)

Wednesday afternoon I went to a panel organized by the Harvard Undergraduate Mathematics Association for a "Gender Gap on Math Discussion". I went both to hear what was going on (as there's still that gender gap in CS, and we're always eager to hear ways that we might do things better that might reduce that), and for moral support for some people I know who were involved. The panel was co-organized by Meena Boppana, an undergraduate who did research with me the summer before last and is currently a star in my graduate class, and one of the panelists was Hilary Finucane, a graduate student who I advised on her senior thesis and collaborated with on multiple papers with when she was an undergraduate at Harvard. I should note that Meena and several others had done a survey of Harvard math undergraduates which had highlighted some issues that would be a starting point for discussion.

I could only stay for the first half or so, but it seemed very positive. A number of faculty showed up, which was promising. My take on the panel's take was was that they were interested in how to make improvements in the culture, and the goal was to try to start figuring out how that could happen, in part by sharing their experiences. The discussion was both balanced and thoughtful, presented positives with negatives, but focused on how to improve things. There's a writeup in the Crimson with more details. The main point that came out in the first half was something I've seen and heard before: the importance of having a community, including (but not necessarily limited to) a community of women that can offer support, guidance, and mentorship, but also just so you don't continually feel like the only woman in the room.

And as long as we're on the subject, there's been a number of recent stories (or older stories where I've recently seen the links) on women in math and computer science. Focusing on Harvard to start, there's a nice writeup about Harvard's Women in Computer Science group, which has helped provide that community that encourages women to take classes in and concentrate in computer science. An article from last year discusses progress at Harvard in closing the gender gap in computer science. There was even an article in the Harvard Political Review covering gender gap issues at Harvard.

Outside of Harvard, from sources on Google+ I've seen a blog post with an interesting slide deck from one Katie Cunningham that provides a great starting point of discussion about the culture and women in computer science. And, finally, a link to something simultaneously sad and funny (things-male-tech-colleagues-have-actually-said-annotated) that reminds us why we have to keep trying to improve the culture.

I could only stay for the first half or so, but it seemed very positive. A number of faculty showed up, which was promising. My take on the panel's take was was that they were interested in how to make improvements in the culture, and the goal was to try to start figuring out how that could happen, in part by sharing their experiences. The discussion was both balanced and thoughtful, presented positives with negatives, but focused on how to improve things. There's a writeup in the Crimson with more details. The main point that came out in the first half was something I've seen and heard before: the importance of having a community, including (but not necessarily limited to) a community of women that can offer support, guidance, and mentorship, but also just so you don't continually feel like the only woman in the room.

And as long as we're on the subject, there's been a number of recent stories (or older stories where I've recently seen the links) on women in math and computer science. Focusing on Harvard to start, there's a nice writeup about Harvard's Women in Computer Science group, which has helped provide that community that encourages women to take classes in and concentrate in computer science. An article from last year discusses progress at Harvard in closing the gender gap in computer science. There was even an article in the Harvard Political Review covering gender gap issues at Harvard.

Outside of Harvard, from sources on Google+ I've seen a blog post with an interesting slide deck from one Katie Cunningham that provides a great starting point of discussion about the culture and women in computer science. And, finally, a link to something simultaneously sad and funny (things-male-tech-colleagues-have-actually-said-annotated) that reminds us why we have to keep trying to improve the culture.

## Thursday, April 02, 2015

### On the Shannon Centennial

I found in my snail mail mailbox my paper copy of the IEEE Information Theory Society Newsletter. First, I was delighted by the news that Michelle Effros (of Cal Tech) is the new President of the IEEE Information Theory Society. Michelle has a long history of service (as well as, it goes without saying, outstanding research) in the information theory community, and is a great selection for the job.

In her opening column, Michelle discusses the importance of letting people outside of their community know what the information theory research community is doing, especially with the Shannon Centennial (April 30, 2016 will be the 100th anniversary of his birth) coming up. The IT Society will be spearheading outreach efforts as part of the Centennial. As Michelle says,

I have always thought that the Information Theory community and the computer science community -- particularly on the theory side -- should interact and communicate more, as there are huge overlaps in the problems being studied and still significant differences in techniques used (although there's more and more crossover in this regard). Perhaps the Shannon Centennial will provide some grand opportunities for the two communities to come together, to promote the Shannon legacy, and as a side benefit to learn more from and about each other.

In her opening column, Michelle discusses the importance of letting people outside of their community know what the information theory research community is doing, especially with the Shannon Centennial (April 30, 2016 will be the 100th anniversary of his birth) coming up. The IT Society will be spearheading outreach efforts as part of the Centennial. As Michelle says,

Every school child learns the name of Albert Einstein; his most famous equation has somehow entered the realm of popular culture. Why is it that so few people know the name or have heard about the contributions of Claude Elwood Shannon?In Computer Science, Turing is our "guiding light", and we had a very successful centenary celebration -- as well as a recent popular movie The Imitation Game -- to make Turing's life and work as well as the importance of computer science as a scientific discipline more well known and understood to the rest of the world. But Shannon, too, is one of the guiding lights of computer science; it is hard to imagine large parts of computer science theory and networking, for example, without the foundations laid out by Shannon in developing the theory of communications.

I have always thought that the Information Theory community and the computer science community -- particularly on the theory side -- should interact and communicate more, as there are huge overlaps in the problems being studied and still significant differences in techniques used (although there's more and more crossover in this regard). Perhaps the Shannon Centennial will provide some grand opportunities for the two communities to come together, to promote the Shannon legacy, and as a side benefit to learn more from and about each other.

## Saturday, March 28, 2015

### Links: HBR article on Women in STEM and AAUW Report

A latest article on bias issues for women in STEM by Harvard Business Review.

A slide presentation by AAUW related to their report: Solving the Equation: The Variables for Women’s Success in Engineering and Computing. The full report can be downloaded for free here.

A slide presentation by AAUW related to their report: Solving the Equation: The Variables for Women’s Success in Engineering and Computing. The full report can be downloaded for free here.

## Wednesday, March 18, 2015

### Double Hashing (Lueker and Molodowitch)

A subject I've grown interested in, related to multiple-choice hashing schemes, is when (and why) double hashing can be used in place of "random hashing" with an asymptotically negligible difference in performance.

One early, useful work on this subject is by Lueker and Molodowitch. They provide a very nice coupling argument between double hashing and random hashing in the setting of open address hashing in their paper More Analysis of Double Hashing. (The original paper appeared in STOC 1988.) In this post I'll summarize their argument. I apologize that both the text and my exposition might be a little rough.

They work in the open address hashing setting; each key runs through a permutation of the table locations when it is being placed, and it placed in the first empty location, with each location holding a single key. When searching for a key, we run sequentially through its permutation; we either eventually find the element or we find an empty slot, in which case we know the key was not in the table, and the search was unsuccessful. We measure the expected time for an unsuccessful search when a table with m slots is loaded with pm keys for a constant fraction p. For convenience we will have m be prime, as this will simplify matters when we consider double hashing. If each key's permutation is completely uniform over all permutations, we call this random hashing, and the expected time to search for key not in the table is 1/(1-p) + o(1); with some work you can get that it is 1/(1-p)+O(1/m), but we will not concern ourselves so much with the low order terms here.

With double hashing, for a key x, the permutation is given by h_1(x)+ j h_2(x) mod m for hash functions h_1 and h_2, where h_1(x) is uniform over the range [0,m-1], h_2(x) is uniform over the range [1,m-1], and the permutation takes the values in the order j=0,1,2,... This gives a permutation (because m is prime), and with double hashing, you just need two random hash values, which from a theoretical standpoint is "much less randomness" than a fully random permutation, and from a practical standpoint is easier to implement.

What Lueker and Molodowitch show is that for any (constant) load factor p, with double hashing, the expected time for an unsuccessful search remains 1/(1-p) + o(1). They show this through a coupling, which shows that double hashing and random hashing can be coupled so the "the same thing happens" -- that is, the key goes into the same slot -- under both double hashing and random hashing most of the time. Unfortunately, it doesn't happen all the time; the coupling is not strong enough to say that all the keys are placed the same with high probability. But they show that they can arrange the coupling so that thing work out nicely just the same.

To start, let us start with a setting where we have loaded our tables with n keys using random hashing, and now take two copies of our state, and consider a single step of random hashing in one copy and double hashing in the other copy, side by side. Clearly, for random hashing, the probability that a key is placed in any empty slot is 1/(m-n) for each slot. In expectation (over the random past), by symmetry, for double hashing the expected probability that a key is placed in any empty is 1/(m-n), but the actual probability for each slot will depend on the configuration. But what they show, using Chernoff bounds, is the the actual probability the key is placed in each slot is at most q/(m-n) for some q that is (1+o(1)), with high probability over the past random placements of the n keys.

Now for the coupling. Starting from empty, at each step we use double hashing in both of our copies with probability 1/q = 1- o(1). Note that this ensures that the probability a key is placed in the "random hashing" copy of the process is at most 1/(m-n), so far. So with probability 1/q, we have placed the key in the same slot in both tables, and so it is as though we've done random hashing for this step.

But what about what happens with probability 1-1/q? Maybe we could ignore it, if 1/q was 1-o(1/n) for example, as a low probability event; unfortunately, that's not the case. In particular, we actually expect that the coupling will fail for some smallish (polylogarithmic) number of steps.

Instead, with probability 1-1/q we place the key so that the step follows random hashing in total. I'm not saying with probability 1-1/q we place the key at random; I'm saying we place the key so that, in total (including the 1-1/q probability first step where they key was placed by double hashing) we place the key so that, overall, the probability any empty slot obtains the key is 1/(m-n). Another way of thinking about this is in the other direction; my coupling always placed the key according the random hashing, and with probability 1/q (which again is very close to 1) that matches what would be done with double hashing.

So in our random hashing copy of the table, we just placed a key according to random hashing. How should we think of what is happening over in the double hashing copy? For that table, with probability 1/q all went fine -- a key was placed by double hashing -- and with probability 1-1/q some key just dropped into the table that wasn't placed by double hashing. It's like an extra present from above. But it's not a key placed by double hashing.

The next part of the argument is to recognize that that's OK, in the following sense. If you simply add a key anywhere is an open addressed hash table, you just make things worse, in a very specific way. Any slot in the table that would have been filled if you hadn't put in that key will still be filled at the end of the process even when you add that key. That is, if S is the set of slots that would contain a key if no extra keys get placed, and S' is the set of slots that contain a key if you, at various points in the process, just add some extra keys anywhere at any point, then a simple induction gives S is a subset S'.

So now let's consider multiple steps of this coupling. At each step, the ball is actually placed according to random hashing, so at every point in the process, the "state" is that of a random hashing process. On the double hashing side of the coupling, with probability 1/q a ball was placed by double hashing, and with probability 1-1/q an extra ball was just placed. So if we count the number of balls placed by double hashing, when we reach the time when n keys have been placed by double hashing in this process, on average n/(1-1/q) = n(1+o(1)) keys (in expectation -- by Chernoff bounds one can get a high probability result) have been placed overall.

The result: placing n keys by double hashing is stochastically dominated (in terms of the keys that have been placed) by placing n(1+o(1)) keys by random hashing. In particular, after we place n=pm keys using double hashing, the expected time for an unsuccessful search is bounded above the expected time for unsuccessful search after putting in pm+o(pm) keys using random hashing, which is 1/(1-p) + o(1). You can do a similar sort of coupling to show that double hashing stochastically dominates placing n(1-o(1)) keys by random hashing. As a result, asymptotically, there's only an o(1) difference in terms of the expected time for unsuccessful search, a result which explains the negligible difference in performance one sees in implementation.

One early, useful work on this subject is by Lueker and Molodowitch. They provide a very nice coupling argument between double hashing and random hashing in the setting of open address hashing in their paper More Analysis of Double Hashing. (The original paper appeared in STOC 1988.) In this post I'll summarize their argument. I apologize that both the text and my exposition might be a little rough.

They work in the open address hashing setting; each key runs through a permutation of the table locations when it is being placed, and it placed in the first empty location, with each location holding a single key. When searching for a key, we run sequentially through its permutation; we either eventually find the element or we find an empty slot, in which case we know the key was not in the table, and the search was unsuccessful. We measure the expected time for an unsuccessful search when a table with m slots is loaded with pm keys for a constant fraction p. For convenience we will have m be prime, as this will simplify matters when we consider double hashing. If each key's permutation is completely uniform over all permutations, we call this random hashing, and the expected time to search for key not in the table is 1/(1-p) + o(1); with some work you can get that it is 1/(1-p)+O(1/m), but we will not concern ourselves so much with the low order terms here.

With double hashing, for a key x, the permutation is given by h_1(x)+ j h_2(x) mod m for hash functions h_1 and h_2, where h_1(x) is uniform over the range [0,m-1], h_2(x) is uniform over the range [1,m-1], and the permutation takes the values in the order j=0,1,2,... This gives a permutation (because m is prime), and with double hashing, you just need two random hash values, which from a theoretical standpoint is "much less randomness" than a fully random permutation, and from a practical standpoint is easier to implement.

What Lueker and Molodowitch show is that for any (constant) load factor p, with double hashing, the expected time for an unsuccessful search remains 1/(1-p) + o(1). They show this through a coupling, which shows that double hashing and random hashing can be coupled so the "the same thing happens" -- that is, the key goes into the same slot -- under both double hashing and random hashing most of the time. Unfortunately, it doesn't happen all the time; the coupling is not strong enough to say that all the keys are placed the same with high probability. But they show that they can arrange the coupling so that thing work out nicely just the same.

To start, let us start with a setting where we have loaded our tables with n keys using random hashing, and now take two copies of our state, and consider a single step of random hashing in one copy and double hashing in the other copy, side by side. Clearly, for random hashing, the probability that a key is placed in any empty slot is 1/(m-n) for each slot. In expectation (over the random past), by symmetry, for double hashing the expected probability that a key is placed in any empty is 1/(m-n), but the actual probability for each slot will depend on the configuration. But what they show, using Chernoff bounds, is the the actual probability the key is placed in each slot is at most q/(m-n) for some q that is (1+o(1)), with high probability over the past random placements of the n keys.

Now for the coupling. Starting from empty, at each step we use double hashing in both of our copies with probability 1/q = 1- o(1). Note that this ensures that the probability a key is placed in the "random hashing" copy of the process is at most 1/(m-n), so far. So with probability 1/q, we have placed the key in the same slot in both tables, and so it is as though we've done random hashing for this step.

But what about what happens with probability 1-1/q? Maybe we could ignore it, if 1/q was 1-o(1/n) for example, as a low probability event; unfortunately, that's not the case. In particular, we actually expect that the coupling will fail for some smallish (polylogarithmic) number of steps.

Instead, with probability 1-1/q we place the key so that the step follows random hashing in total. I'm not saying with probability 1-1/q we place the key at random; I'm saying we place the key so that, in total (including the 1-1/q probability first step where they key was placed by double hashing) we place the key so that, overall, the probability any empty slot obtains the key is 1/(m-n). Another way of thinking about this is in the other direction; my coupling always placed the key according the random hashing, and with probability 1/q (which again is very close to 1) that matches what would be done with double hashing.

So in our random hashing copy of the table, we just placed a key according to random hashing. How should we think of what is happening over in the double hashing copy? For that table, with probability 1/q all went fine -- a key was placed by double hashing -- and with probability 1-1/q some key just dropped into the table that wasn't placed by double hashing. It's like an extra present from above. But it's not a key placed by double hashing.

The next part of the argument is to recognize that that's OK, in the following sense. If you simply add a key anywhere is an open addressed hash table, you just make things worse, in a very specific way. Any slot in the table that would have been filled if you hadn't put in that key will still be filled at the end of the process even when you add that key. That is, if S is the set of slots that would contain a key if no extra keys get placed, and S' is the set of slots that contain a key if you, at various points in the process, just add some extra keys anywhere at any point, then a simple induction gives S is a subset S'.

So now let's consider multiple steps of this coupling. At each step, the ball is actually placed according to random hashing, so at every point in the process, the "state" is that of a random hashing process. On the double hashing side of the coupling, with probability 1/q a ball was placed by double hashing, and with probability 1-1/q an extra ball was just placed. So if we count the number of balls placed by double hashing, when we reach the time when n keys have been placed by double hashing in this process, on average n/(1-1/q) = n(1+o(1)) keys (in expectation -- by Chernoff bounds one can get a high probability result) have been placed overall.

The result: placing n keys by double hashing is stochastically dominated (in terms of the keys that have been placed) by placing n(1+o(1)) keys by random hashing. In particular, after we place n=pm keys using double hashing, the expected time for an unsuccessful search is bounded above the expected time for unsuccessful search after putting in pm+o(pm) keys using random hashing, which is 1/(1-p) + o(1). You can do a similar sort of coupling to show that double hashing stochastically dominates placing n(1-o(1)) keys by random hashing. As a result, asymptotically, there's only an o(1) difference in terms of the expected time for unsuccessful search, a result which explains the negligible difference in performance one sees in implementation.

## Monday, March 16, 2015

### Power of Randomness at Georgia Tech

I'm spending (part of) the week at "The Power of Randomness in Computation Workshop", an IMA (Institute for Mathematics and its Applications) and ARC (Georgia Tech Algorithm and Randomness Center) co-sponsored workshop at Georgia Tech. Here's the schedule. I'm told slides will eventually be put up somewhere on the IMA website for such things. Great organization at Georgia Tech -- a big crowd in a very nice room, lots of food and coffee, all very well organized. They even had Ben Affleck waiting in front of the building for us this morning. He seemed to be a little busy shooting a movie to greet us properly, but maybe he'll have a bit more time to chat tomorrow.

Besides Ben, a few other highlights:

Leslie Goldberg started things of talking about the complexity of approximating complex-valued Ising and Tutte partition functions. I remember the Ising/Tutte models (mostly from graduate school and shortly after); now there are connections between various problems in quantum computing and these functions on complex values, which (or course) I had not known.

Nike Sun gave a talk on the exact k-SAT threshold (for large k). It was very clearly presented and gave the argument at the intuitive level. I gained some insight into why the "locally random tree" type argument I've enjoyed in coding/belief propagation arguments breaks down in certain satisfiability problems, due to clustering of solutions and other challenging correlations, and how those issues can be handled. I started to understand (I think) the point of replica symmetry breaking arguments and how they were used to guide the analysis of the k-SAT problem.

Other talks from the day: Amin-Coja Oghlan also talked about replica symmetry techniques and their uses for random graph coloring problems, Eli Upfal talked about some new shuffling techniques for oblivious storage dubbed the Melbourne shuffle, Aravind Srinivasan gave a talk on the Lovasz Local Lemma (staring from the Moser-Tardos results and showing how these arguments carry forward and give greater power and insight into the use of the LLL for additional problems), and I talked about invertible Bloom lookup tables and briefly mentioned a few other unrelated things in progress.

Besides Ben, a few other highlights:

Leslie Goldberg started things of talking about the complexity of approximating complex-valued Ising and Tutte partition functions. I remember the Ising/Tutte models (mostly from graduate school and shortly after); now there are connections between various problems in quantum computing and these functions on complex values, which (or course) I had not known.

Nike Sun gave a talk on the exact k-SAT threshold (for large k). It was very clearly presented and gave the argument at the intuitive level. I gained some insight into why the "locally random tree" type argument I've enjoyed in coding/belief propagation arguments breaks down in certain satisfiability problems, due to clustering of solutions and other challenging correlations, and how those issues can be handled. I started to understand (I think) the point of replica symmetry breaking arguments and how they were used to guide the analysis of the k-SAT problem.

Other talks from the day: Amin-Coja Oghlan also talked about replica symmetry techniques and their uses for random graph coloring problems, Eli Upfal talked about some new shuffling techniques for oblivious storage dubbed the Melbourne shuffle, Aravind Srinivasan gave a talk on the Lovasz Local Lemma (staring from the Moser-Tardos results and showing how these arguments carry forward and give greater power and insight into the use of the LLL for additional problems), and I talked about invertible Bloom lookup tables and briefly mentioned a few other unrelated things in progress.

## Tuesday, March 03, 2015

### Hate EasyChair

I just typed in a nice long review on EasyChair. Yes, I prefer doing this with the online form when I'm just sitting around and have time to do a review.

Apparently I didn't hit one of the score buttons (although I'm pretty sure I did, let's give EasyChair the benefit of the doubt there) so EasyChair says there's an error and, of course, forgets my nice long typed review when it takes me back to the review page, so I'll get to re-do and re-type it later.

Sigh. I guess I'll go back to doing my reviews in a text file and cutting and pasting. No, this has not happened to me in recent memory in HotCRP. Put this down as one more reason (but not the only one) why I don't like EasyChair and would prefer a better designed system (like HotCRP...).

Apparently I didn't hit one of the score buttons (although I'm pretty sure I did, let's give EasyChair the benefit of the doubt there) so EasyChair says there's an error and, of course, forgets my nice long typed review when it takes me back to the review page, so I'll get to re-do and re-type it later.

Sigh. I guess I'll go back to doing my reviews in a text file and cutting and pasting. No, this has not happened to me in recent memory in HotCRP. Put this down as one more reason (but not the only one) why I don't like EasyChair and would prefer a better designed system (like HotCRP...).

## Wednesday, February 11, 2015

### New Heapable Subsequence Paper

In the "only a dozen people could care about this category"...

About 4 1/2 years ago, I posted about a paper we had put up on the arxiv about Heapable Sequences and Subsequences. The basic combinatorial structure we were looking at is a seemingly natural generalization of the idea of Longest Increasing Subsequences. Say that a sequence is heapable if you can sequentially place the items into a (binary, increasing) heap, so each new item is the child of some item already in the heap. So, for example, 1 4 2 3 5 is heapable, but 1 5 3 4 2 is not. Once you have this idea, you can ask about things like the Longest Heapable Subsequence of a sequence (algorithms for it, expected length with a random permutation, etc.). Our paper had some results and lots of open questions.

I admit, when we did this paper I was hoping that some combinatorialist(s) would find the notion compelling, take up the questions, and find some cool connections. Longest Increasing Subsequences are somehow related to Young tableaux, interacting particle systems, and all sorts of other cool things. So what about Longest Heapable Subsequences?

I had to wait a few years, but Gabriel Istrate and Cosmin Bonchis recently put a paper up on the arxiv that makes these connections. Here's the abstract:

I love the new conjecture that the expected minimal number of heapable subsequences a random sequence decomposes into is ((1+sqrt{5})/2) ln n. (It's clearly at least ln n, the expected number of minima in the sequence.)

There are still all sort of open questions, that seem surprisingly difficult; and I certainly can't claim I know of any important practical applications. But Longest Heapable Subsequences just appeal to me as a simple, straightforward mathematical object that I wish I understood more.

For simple-sounding but apparently difficult open questions, as far as I know, the answer to even the basic question of "What is the formula for how many sequences of length n are heapable?" is still not known. Similarly, I think the question of finding an efficient algorithm for determining the Longest Heapable Subsequence (or showing it is hard for some class) is open as well.

About 4 1/2 years ago, I posted about a paper we had put up on the arxiv about Heapable Sequences and Subsequences. The basic combinatorial structure we were looking at is a seemingly natural generalization of the idea of Longest Increasing Subsequences. Say that a sequence is heapable if you can sequentially place the items into a (binary, increasing) heap, so each new item is the child of some item already in the heap. So, for example, 1 4 2 3 5 is heapable, but 1 5 3 4 2 is not. Once you have this idea, you can ask about things like the Longest Heapable Subsequence of a sequence (algorithms for it, expected length with a random permutation, etc.). Our paper had some results and lots of open questions.

I admit, when we did this paper I was hoping that some combinatorialist(s) would find the notion compelling, take up the questions, and find some cool connections. Longest Increasing Subsequences are somehow related to Young tableaux, interacting particle systems, and all sorts of other cool things. So what about Longest Heapable Subsequences?

I had to wait a few years, but Gabriel Istrate and Cosmin Bonchis recently put a paper up on the arxiv that makes these connections. Here's the abstract:

We investigate partitioning of integer sequences into heapable subsequences (previously defined and established by Mitzenmacher et al). We show that an extension of patience sorting computes the decomposition into a minimal number of heapable subsequences (MHS). We connect this parameter to an interactive particle system, a multiset extension of Hammersley's process, and investigate its expected value on a random permutation. In contrast with the (well studied) case of the longest increasing subsequence, we bring experimental evidence that the correct asymptotic scaling is(Note, that should really be "Byers et al...")1+5√2⋅ln(n) . Finally we give a heap-based extension of Young tableaux, prove a hook inequality and an extension of the Robinson-Schensted correspondence.

I love the new conjecture that the expected minimal number of heapable subsequences a random sequence decomposes into is ((1+sqrt{5})/2) ln n. (It's clearly at least ln n, the expected number of minima in the sequence.)

There are still all sort of open questions, that seem surprisingly difficult; and I certainly can't claim I know of any important practical applications. But Longest Heapable Subsequences just appeal to me as a simple, straightforward mathematical object that I wish I understood more.

For simple-sounding but apparently difficult open questions, as far as I know, the answer to even the basic question of "What is the formula for how many sequences of length n are heapable?" is still not known. Similarly, I think the question of finding an efficient algorithm for determining the Longest Heapable Subsequence (or showing it is hard for some class) is open as well.

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