(1 + )-Approximate shortest paths in dynamic streams

Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of 2−1 versus n1+1/, for an integer parameter . (In fact, existing solutions also incur an extra factor of 1 + in the stretch for weighted graphs, and an additional factor of logO(1) n in the space.) The only existing solution of the second type uses n1/2−O(1/) passes over the stream (for space O(n1+1/)), and applies only to unweighted graphs. In this paper we show that (1+)-approximate single-source shortest paths can be computed with ˜O (n1+1/) space using just constantly many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs. Moreover, the same result applies for multi-source shortest paths, as long as the number of sources is O(n1/). We achieve these results by devising efficient dynamic streaming constructions of (1 + , )-spanners and hopsets. On our way to these results, we also devise a new dynamic streaming algorithm for the 1-sparse recovery problem. Even though our algorithm for this task is slightly inferior to the existing algorithms of [26, 11], we believe that it is of independent interest. 2012 ACM Subject Classification Theory of computation ! Streaming models; Theory of computation ! Streaming, sublinear and near linear time algorithms; Theory of computation ! Shortest paths; Theory of computation ! Sparsification and spanner

Brief announcement: (1+)-approximate shortest paths in dynamic streams

Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of 2k - 1 versus n1+1/k , for an integer parameter k. (In fact, existing solutions also incur an extra factor of 1+ in the stretch for weighted graphs, and an additional factor of logO(1) n in the space.) The only existing solution of the second type uses n1/2-O(1/k) passes over the stream (for space O(n1+1/k )), and applies only to unweighted graphs

Processing massive graphs under limited visibility

Graphs are one of the most important and widely used combinatorial structures in mathematics. Their ability to model many real world scenarios which involve a large network of related entities make them useful across disciplines. They are useful as an abstraction in the analysis of networked structures such as the Internet, social networks, road networks, biological networks and many more. The graphs arising out of many real world phenomenon can be very large and they keep evolving over time. For example, Facebook reported over 2:9 billion monthly active users in 2022. Another very large and dynamic network is the human brain consisting of around 1011 nodes and many more edges. These large and evolving graphs present new challenges for algorithm designers. Traditional graph algorithms designed to work with centralised and sequential computing models are rendered useless due to their prohibitively high resource usage. In fact one needs huge amounts of resources just to read the entire graph. A number of new theoretical models have been devised over the years to keep up with the trends in the modern computing systems capable of handing massive input datasets. Some of these models such as streaming model and the query model allow the algorithm to view the graph piecemeal. In some cases, the model allows the graph to be processed by a set of interconnected computing elements such as in distributed computing. In this thesis we address some graph problems in these non-centralised, non-sequential models of computing with a limited access to the input graph. Specifically, we address three different graph problems, each in a different computing model. The first problem we look at is the computation of approximate shortest paths in dynamic streams. The second problem deals with finding kings in tournament graphs, given query access to the arcs of the tournament. The third and the final problem we investigate is a local test criteria for testing the expansion of a graph in the distributed CONGEST model

When You Come at the King You Best Not Miss

A tournament is an orientation of a complete graph. We say that a vertex x in a tournament T controls another vertex y if there exists a directed path of length at most two from x to y. A vertex is called a king if it controls every vertex of the tournament. It is well known that every tournament has a king. We follow Shen, Sheng, and Wu [Jian Shen et al., 2003] in investigating the query complexity of finding a king, that is, the number of arcs in T one has to know in order to surely identify at least one vertex as a king. The aforementioned authors showed that one always has to query at least ?(n^{4/3}) arcs and provided a strategy that queries at most O(n^{3/2}). While this upper bound has not yet been improved for the original problem, [Biswas et al., 2017] proved that with O(n^{4/3}) queries one can identify a semi-king, meaning a vertex which controls at least half of all vertices. Our contribution is a novel strategy which improves upon the number of controlled vertices: using O(n^{4/3} polylog n) queries, we can identify a (1/2+2/17)-king. To achieve this goal we use a novel structural result for tournaments

Energy Optimization of Memory Intensive Parallel Workloads

Energy consumption is an important concern in modern multicore processors. The energy consumed during the execution of an application can be minimized by tuning the hardware state utilizing knobs such as frequency, voltage etc. The existing theoretical work on energy mini- mization using Global DVFS (Dynamic Voltage and Frequency Scaling), despite being thorough, ignores the energy consumed by the CPU on memory accesses and the dynamic energy consumed by the idle cores. This article presents an analytical model for the performance and the overall energy consumed by the CPU chip on CPU instructions as well as the memory accesses without ignoring the dynamic energy consumed by the idle cores. We present an analytical framework around our energy-performance model to predict the operating frequencies for global DVFS that minimize the overall CPU energy consumption within a performance budget. Finally, we suggest a scheduling criteria for energy aware scheduling of memory intensive parallel applications

When you come at the king you best not miss

A tournament is an orientation of a complete graph. We say that a vertex x in a tournament ⃗ T controls another vertex y if there exists a directed path of length at most two from x to y. A vertex is called a king if it controls every vertex of the tournament. It is well known that every tournament has a king. We follow Shen, Sheng, and Wu [ 8] in investigating the query complexity of finding a king, that is, the number of arcs in ⃗ T one has to know in order to surely identify at least one vertex as a king. The aforementioned authors showed that one always has to query at least Ω(n4/3) arcs and provided a strategy that queries at most O(n3/2). While this upper bound has not yet been improved for the original problem, Biswas et al. [3] proved that with O(n4/3) queries one can identify a semi-king, meaning a vertex which controls at least half of all vertices. Our contribution is a novel strategy which improves upon the number of controlled vertices: using O(n4/3 polylog n) queries, we can identify a ( 1/2 + 2/17 )-king. To achieve this goal we use a novel structural result for tournaments