Consider the free algebra An generated over Q by n generators x1, . . . , xn. Interesting objects attached to A = An are members of its lower central series, Li = Li(A), defined inductively by L1 = A, Li+1 = [A, Li], and their associated graded components Bi = Bi(A) defined as Bi = Li/Li+1. These quotients Bi for i ≥ 2, as well as the reduced quotientB1 = A/(L2 + AL3), exhibit a rich geometric structure, as shown by Feigin and Shoikhet [FS] and later authors,We study the same problem over the integers Z and finite fields Fp. New phenomena arise, namely, torsion in Bi over Z, and jumps in dimension over Fp. We describe the torsion in the reduced quotientB1 and B2 geometrically in terms of the De Rham cohomology of Z n . As a corollary we obtain a complete description ofB1(An(Z)) andB1(An(Fp)), as well as of B2(An(Z[1/2])) and B2(An(Fp)), p > 2. We also give theoretical and experimental results for Bi with i > 2, formulating a number of conjectures and questions on their basis. Finally, we discuss the supercase, when some of the generators are odd and some are even, and provide some theoretical results and experimental data in this case.

The single-and multi-processor cup games can be used to model natural problems in areas such as processor scheduling, deamortization, and buffer management.At the beginning of the single-processor cup game, n cups sit in a row, initially empty. In each step of the game, a filler distributes 1 unit of water among the cups, and then an emptier selects a cup and removes 1 + ε units from that cup. The goal of the emptier is to minimize the amount of water in the fullest cup, also known as the backlog. It is known that the greedy algorithm (i.e., empty the fullest cup) achieves backlog O(log n), and that no deterministic algorithm can do better.We show that the performance of the greedy algorithm can be greatly improved with a small amount of randomization: After any step i, and for any k ≥ Ω(log ε −1 ), the emptier achieves backlog at most O(k) with probability at least 1 − O(2 −2 k ). Our algorithm, which we call the smoothed greedy algorithm, can also be interpreted as a one-shot smoothed analysis of the standard greedy algorithm.Whereas bounds for the single-processor cup game have been known for more than fifteen years, proving nontrivial bounds on backlog for the multi-processor extension has remained open. We present a simple analysis of the greedy algorithm for the multi-processor cup game, establishing a backlog of O(ε −1 log n), as long as δ, the game's other speed-augmentation constant, is at least 1/poly(n).Turning to randomized algorithms, we encounter an unexpected phenomenon: When the number of processors p is large, the backlog after each step drops to constant with large probability. Specifically, we show that if δ and ε satisfy reasonable constraints, then there exists an algorithm that bounds the backlog after a given step by three or less with probability at least 1 − O(exp(−Ω(ε 2 p)). We further extend the guarantees of our randomized algorithm to consider larger backlogs.When ε is constant, we prove that our results are asymptotically optimal, in the sense that no algorithms can achieve better bounds, up to constant factors in the backlog and in p. Moreover, we prove robustness results, demonstrating that our randomized algorithms continue to behave well even when placed in bad starting states.The small amount of resource augmentation used by the smoothed greedy algorithm makes it robust to the setting in which cups begin in a bad initial starting state. In particular, we show that if b units of water are maliciously placed into cups at the beginning of the game, then for steps i > b ε , the b units of water have no affect on the guarantees given by the algorithm.The multi-processor cup game. The multi-processor version of the same scheduling question has proven to be much harder. In each step of the multi-processor cup game, the filler distributes some amount of water proportional to p, the number of processors, among the cups (i.e., threads), and the emptier picks p cups and removes a unit of water from each. If the filler is unrestricted in their placement of the water, then they can ensure an...

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We study a family of equivalence relations on S n , the group of permutations on n letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of S c .When the partition is of S 3 and has one nontrivial part and that part is of size greater than two, we provide formulas for the number of classes created in each previously unsolved case. When the partition is of S 3 and has two nontrivial parts, each of size two (as do the Knuth and forgotten relations), we enumerate the classes for 13 of the 14 unresolved cases. In two of these cases, enumerations arise which are the same as those yielded by the Knuth and forgotten relations. The reasons for this phenomenon are still largely a mystery.Lemma 2.17. Every permutation not ending with 1 is equivalent to a permutation of the form . . . 1j for some j under the {123, 132, 312}-equivalence.Proof. Such a permutation is reached through repeated applications of 123 → 312 and 132 → 312 using the actual letter 1.

In each step of the p-processor cup game on n cups, a filler distributes up to p units of water among the cups, subject only to the constraint that no cup receives more than 1 unit of water; an emptier then removes up to 1 unit of water from each of p cups. Designing strategies for the emptier that minimize backlog (i.e., the height of the fullest cup) is important for applications in processor scheduling, buffer management in networks, quality of service guarantees, and deamortization.We prove that the greedy algorithm (i.e., the emptyfrom-fullest-cups algorithm) achieves backlog O(log n) for any p ≥ 1. This resolves a long-standing open problem for p > 1, and is asymptotically optimal as long as n ≥ 2p.If the filler is an oblivious adversary, then we prove that there is a randomized emptying algorithm that achieve backlog O(log p + log log n) with probability 1 − 2 − polylog(n) for 2 polylog(n) steps. This is known to be asymptotically optimal when n is sufficiently large relative to p. The analysis of the randomized algorithm can also be reinterpreted as a smoothed analysis of the deterministic greedy algorithm.Previously, the only known bound on backlog for p > 1, and the only known randomized guarantees for any p (including when p = 1), required the use of resource augmentation, meaning that the filler can only distribute at most p(1 − ǫ) units of water in each step, and that the emptier is then permitted to remove 1 + δ units of water from each of p cups, for some ǫ, δ > 0. * MIT CSAIL.

We present an algorithm for approximating the edit distance ed(x, y) between two strings x and y in time parameterized by the degree to which one of the strings x satisfies a natural pseudorandomness property. The pseudorandomness model is asymmetric in that no requirements are placed on the second string y, which may be constructed by an adversary with full knowledge of x.We say that x is (p, B)-pseudorandom if all pairs a and b of disjoint B-letter substrings of x satisfy ed(a, b) ≥ pB.

We study a family of equivalence relations on S n , the group of permutations on n letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of S c . In particular, we are interested in the number of classes created in S n by each relation and in characterizing these classes.Imposing the condition that the partition of S c has one nontrivial part containing the cyclic shifts of a single permutation, we find enumerations for the number of nontrivial classes. When the permutation is the identity, we are able to compare the sizes of these classes and connect parts of the problem to Young tableaux and Catalan lattice paths.Imposing the condition that the partition has one nontrivial part containing all of the permutations in S c beginning with 1, we both enumerate and characterize the classes in S n . We do the same for the partition that has two nontrivial parts, one containing all of the permutations in S c beginning with 1, and one containing all of the permutations in S c ending with 1.

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