Combinatorial algorithms for web search engines: three success stories

How much can smart combinatorial algorithms improve web search engines? To address this question we will describe three algorithms that have had a positive impact on web search engines: The PageRank algorithm, algorithms for finding near-duplicate web pages, and algorithms for index server loadbalancing

Fine-Grained Complexity Lower Bounds for Families of Dynamic Graphs

A dynamic graph algorithm is a data structure that answers queries about a property of the current graph while supporting graph modifications such as edge insertions and deletions. Prior work has shown strong conditional lower bounds for general dynamic graphs, yet graph families that arise in practice often exhibit structural properties that the existing lower bound constructions do not possess. We study three specific graph families that are ubiquitous, namely constant-degree graphs, power-law graphs, and expander graphs, and give the first conditional lower bounds for them. Our results show that even when restricting our attention to one of these graph classes, any algorithm for fundamental graph problems such as distance computation or approximation or maximum matching, cannot simultaneously achieve a sub-polynomial update time and query time. For example, we show that the same lower bounds as for general graphs hold for maximum matching and (s,t)-distance in constant-degree graphs, power-law graphs or expanders. Namely, in an m-edge graph, there exists no dynamic algorithms with both O(m^{1/2 - ?}) update time and O(m^{1 -?}) query time, for any small ? > 0. Note that for (s,t)-distance the trivial dynamic algorithm achieves an almost matching upper bound of constant update time and O(m) query time. We prove similar bounds for the other graph families and for other fundamental problems such as densest subgraph detection and perfect matching

Finding related pages in the World Wide Web

When using traditional search engines, users have to formulate queries to describe their information need. This paper discusses a different approach to Web searching where the input to the search process is not a set of query terms, but instead is the URL of a page, and the output is a set of related Web pages. A related Web page is one that addresses the same topic as the original page. For example, www.washingtonpost.com is a page related to www.nytimes.com, since both are online newspapers. We describe two algorithms to identify related Web pages. These algorithms use only the connectivity information in the Web (i.e., the links between pages) and not the content of pages or usage information. We have implemented both algorithms and measured their runtime performance. To evaluate the effectiveness of our algorithms, we performed a user study comparing our algorithms with Netscape's `What's Related' service (http://home. netscape, com/escapes/related/). Our study showed that the precision at 10 for our two algorithms are 73% better and 51% better than that of Netscape, despite the fact that Netscape uses both content and usage pattern information in addition to connectivity information

Scheduling multicasts on unit-capacity trees and meshes

This paper studies the multicast routing and admission control problem on unit-capacity tree and mesh topologies in the throughput model. The problem is a generalization of the edge-disjoint paths problem and is NP-hard both on trees and meshes. We study both the offline and the online version of the problem: In the offline setting, we give the first constant-factor approximation algorithm for trees, and an O((log log n)2)-factor approximation algorithm for meshes. In the online setting, we give the first polylogarithmic competitive online algorithm for tree and mesh topologies. No polylogarithmic-competitive algorithm is possible on general network topologies (Lower bounds for on-line graph problems with application to on-line circuits and optical routing, in: Proceedings of the 28th ACM Symposium on Theory of Computing, 1996, pp. 531-540) and there exists a polygarithmic lower bound on the competitive ratio of any online algorithm on tree topologies (Making commitments in the face of uncertainity: how to pick a winner almost every time, in: Proceedings of the 28th Annual ACM Symposium on Theory of Computing, 1996, pp. 519-530). We prove the same lower bound for meshes

Average-case analysis of dynamic graph algorithms

We present a model for edge updates with restricted randomness in dynamic graph algorithms and a general technique for analyzing the expected running time of an update operation. This model is able to capture the average case in many applications, since (1) it allows restrictions on the set of edges which can be used for insertions and (2) the type (insertion or deletion) of each update operation is arbitrary, i.e., not random. We use our technique to analyze existing and new dynamic algorithms for the following problems: maximum cardinality matching, minimum spanning forest, connectivity, 2- edge connectivity, k-edge connectivity, k-vertex connectivity, and bipartiteness. Given a random graph G with m0 edges and n vertices and a sequence of l update operations such that the graph contains mi edges after operation i, the expected time for performing the updates for any l is O(l log(n) + sum(i=1 to l) n/sqrt(m_i)) in the case of minimum spanning forests, connectivity, 2-edge connectivity, and bipartiteness. The expected time per update operation is O(n) in the case of maximum matching. We also give improved bounds for k-edge and k-vertex connectivity. Additionally we give an insertions-only algorithm for maximum cardinality matching with worst- case O(n) amortized time per insertion