Thatchaphol Saranurak

Dynamic Minimum Spanning Forest with Subpolynomial Worst-Case Update Time

We present a Las Vegas algorithm for dynamically maintaining a minimum spanning forest of an n-node graph undergoing edge insertions and deletions. Our algorithm guarantees an O(n^{o(1)})} worst-case} update time with high probability. This significantly improves the two recent Las Vegas algorithms by Wulff-Nilsen \cite{Wulff-Nilsen16a} with update time O(n^{0.5-≥ilon}) for some constant ≥ilon 0 and, independently, by Nanongkai and Saranurak \cite{NanongkaiS16} with update time O(n^{0.494}) (the latter works only for maintaining a spanning forest).Our result is obtained by identifying the common framework that both two previous algorithms rely on, and then improve and combine the ideas from both works. There are two main algorithmic components of the framework that are newly improved and critical for obtaining our result. First, we improve the update time from O(n^{0.5-≥ilon}) in \cite{Wulff-Nilsen16a} to O(n^{o(1)}) for decrementally removing all low-conductance cuts in an expander undergoing edge deletions. Second, by revisiting the contraction technique by Henzinger and King \cite{HenzingerK97b} and Holm et al. \cite{HolmLT01, we show a new approach for maintaining a minimum spanning forest in connected graphs with very few (at most (1+o(1))n) edges. This significantly improves the previous approach in \cite{Wulff-Nilsen16a, NanongkaiS16} which is based on Fredericksons 2-dimensional topology tree \cite{Frederickson85} and illustrates a new application to this old technique.

Distributed Exact Weighted All-Pairs Shortest Paths in Õ(n^{5/4}) Rounds

We study computing all-pairs shortest paths (APSP) on distributed networks (the CONGEST model). The goal is for every node in the (weighted) network to know the distance from every other node using communication. The problem admits (1+o(1))-approximation Õ(n)-time algorithms [2], [3], which are matched with \tilde Ω(n)-time lower bounds [4], [5],\footnote{\tilde \Theta, Õ and \tilde Ω hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the weighted case with polynomial approximation ratios.}. No Ω(n) lower bound or o(m) upper bound were known for exact computation.In this paper, we present an Õ(n^{5/4})-time randomized (Las Vegas) algorithm for exact weighted APSP; this provides the first improvement over the naive O(m)-time algorithm when the network is not so sparse. Our result also holds for the case where edge weights are asymmetric} (a.k.a. the directed case where communication is bidirectional). Our techniques also yield an Õ(n^{3/4}k^{1/2}+n)-time algorithm for the k-source shortest paths} problem where we want every node to know distances from k sources; this improves Elkins recent bound [6] when k=\tilde Ω(n^{1/4}).We achieve the above results by developing distributed algorithms on top of the classic scaling technique, which we believe is used for the first time for distributed shortest paths computation. One new algorithm which might be of an independent interest is for the reversed r-sink shortest paths} problem, where we want every of r sinks to know its distances from all other nodes, given that every node already knows its distance to every sink. We show an Õ(n√{r})-time algorithm for this problem. Another new algorithm is called short range extension, where we show that in Õ(n√{h}) time the knowledge about distances can be extended for additional h hops. For this, we use weight rounding to introduce small additive} errors which can be later fixed.

Self-Adjusting Binary Search Trees: What Makes Them Tick?

Splay trees (Sleator and Tarjan [11]) satisfy the so-called access lemma. Many of the nice properties of splay trees follow from it. What makes self-adjusting binary search trees (BSTs) satisfy the access lemma? After each access, self-adjusting BSTs replace the search path by a tree on the same set of nodes (the after-tree). We identify two simple combinatorial properties of the search path and the after-tree that imply the access lemma. Our main result

Greedy Is an Almost Optimal Deque

In this paper we extend the geometric binary search tree (BST) model of Demaine, Harmon, Iacono, Kane, and Pǎtrascu (DHIKP) to accommodate for insertions and deletions. Within this extended model, we study the online Greedy BST algorithm introduced by DHIKP. Greedy BST is known to be equivalent to a maximally greedy (but inherently offline) algorithm introduced independently by Lucas in 1988 and Munro in 2000, conjectured to be dynamically optimal.

Pattern-Avoiding Access in Binary Search Trees

The dynamic optimality conjecture is perhaps the most fundamental open question about binary search trees (BST). It postulates the existence of an asymptotically optimal online BST, i.e. one that is constant factor competitive with any BST on any input access sequence. The two main candidates for dynamic optimality in the literature are splay trees [Sleator and Tarjan, 1985], and GREEDY [Lucas, 1988; Munro, 2000; Demaine et al. 2009]. Despite BSTs being among the simplest data structures in computer science, and despite extensive effort over the past three decades, the conjecture remains elusive. Dynamic optimality is trivial for almost all sequences: the optimum access cost of most length-τ sequences is Θ(n log n), achievable by any balanced BST. Thus, the obvious missing step towards the conjecture is an understanding of the “easy” access sequences, and indeed the most fruitful research direction so far has been the study of specific sequences, whose “easiness” is captured by a parameter of interest. For instance, splay provably achieves the bound of O(nd) when d roughly measures the distances between consecutive accesses (dynamic finger), the average entropy (static optimality), or the delays between multiple accesses of an element (working set). The difficulty of proving dynamic optimality is witnessed by other highly restricted special cases that remain unresolved; one prominent example is the traversal conjecture [Sleator and Tarjan, 1985], which states that preorder sequences (whose optimum is linear) are linear-time accessed by splay trees; no online BST is known to satisfy this conjecture. In this paper, we prove two different relaxations of the traversal conjecture for GREEDY: (i) GREEDY is almost linear for preorder traversal, (ii) if a linear-time preprocessing1 is allowed, GREEDY is in fact linear. These statements are corollaries of our more general results that express the complexity of access sequences in terms of a pattern avoidance parameter k. Pattern avoidance is a well-established concept in combinatorics, and the classes of input sequences thus defined are rich, e.g. the k = 3 case includes preorder sequences. For any sequence X with parameter k, our most general result shows that GREEDY achieves the cost n2α(n)O(k) where α is the inverse Ackermann function. Furthermore, a broad subclass of parameter-k sequences has a natural combinatorial interpretation as k-decomposable sequences. For this class of inputs, we obtain an n2O(k2) bound for GREEDY when preprocessing is allowed. For k = 3, these results imply (i) and (ii). To our knowledge, these are the first upper bounds for GREEDY that are not known to hold for any other online BST. To obtain these results we identify an input-revealing property of GREEDY. Informally, this means that the execution log partially reveals the structure of the access sequence. This property facilitates the use of rich technical tools from forbidden submatrix theory. Further studying the intrinsic complexity of k-decomposable sequences, we make several observations. First, in order to obtain an offline optimal BST, it is enough to bound GREEDY on non-decomposable access sequences. Furthermore, we show that the optimal cost for k-decomposable sequences is Θ(n log k), which is well below the proven performance of all known BST algorithms. Hence, sequences in this class can be seen as a “candidate counterexample” to dynamic optimality.

Smooth heaps and a dual view of self-adjusting data structures

We present a new connection between self-adjusting binary search trees (BSTs) and heaps, two fundamental, extensively studied, and practically relevant families of data structures (Allen,Munro, 1978; Sleator, Tarjan, 1983; Fredman, Sedgewick, Sleator, Tarjan, 1986; Wilber, 1989; Fredman, 1999; Iacono, Özkan, 2014). Roughly speaking, we map an arbitrary heap algorithm within a broad and natural model, to a corresponding BST algorithm with the same cost on a dual sequence of operations (i.e. the same sequence with the roles of time and key-space switched). This is the first general transformation between the two families of data structures. There is a rich theory of dynamic optimality for BSTs (i.e. the theory of competitiveness between BST algorithms). The lack of an analogous theory for heaps has been noted in the literature (e.g. Pettie; 2005, 2008). Through our connection, we transfer all instance-specific lower bounds known for BSTs to a general model of heaps, initiating a theory of dynamic optimality for heaps. On the algorithmic side, we obtain a new, simple and efficient heap algorithm, which we call the smooth heap. We show the smooth heap to be the heap-counterpart of Greedy, the BST algorithm with the strongest proven and conjectured properties from the literature, conjectured to be instance-optimal (Lucas, 1988; Munro, 2000; Demaine et al., 2009). Assuming the optimality of Greedy, the smooth heap is also optimal within our model of heap algorithms. Intriguingly, the smooth heap, although derived from a non-practical BST algorithm, is simple and easy to implement (e.g. it stores no auxiliary data besides the keys and tree pointers). It can be seen as a variation on the popular pairing heap data structure, extending it with a “power-of-two-choices” type of heuristic. For the smooth heap we obtain instance-specific upper bounds, with applications in adaptive sorting, and we see it as a promising candidate for the long-standing question of a simpler alternative to Fibonacci heaps.