STAR: Steiner-Tree Approximation in Relationship Graphs Presented By: Moamen Khet The Hebrew University Of Jerusalem Seminar on Databases & The Internet.

STAR: Steiner-Tree Approximation in Relationship Graphs Presented By: Moamen Khet The Hebrew University Of Jerusalem Seminar on Databases & The Internet 28.12.2010 Presented By: Moamen Khie

Outline Introduction Related Work The STAR Algorithm Approximation Guarantee Time Complexity Approximate Top-K Interconnections Evaluation Conclusion 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Introduction Many modern applications need to deal with graph-based knowledge representation. Such as: biological networks, social online communities, knowledge bases, ER graphs over relational data or Web-extracted entities and many more. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Introduction The data of such graphs exhibits semantics- bearing labels for nodes and edges and can thus be seen as a semantic graph, with nodes corresponding to entities and edge weights capturing the strengths of semantic relationships. Often, this graphs too large to fit into main memory, so querying and analyzing them in efficient way becomes non-trivial. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

YAGO KB For example: The YAGO Knowledge Base. Has been constructed by systematically harvesting semi structured elements (e.g., infoboxes, categories, lists) from Wikipedia. http://www.mpi-inf.mpg.de/yago-naga/yago/ 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

IMDB Another well-know graph with a simpler structure is the IMDB movie database. With movies, actors, producers and other entities as nodes. Movie cast, information about directors, producers... as edges. http://www.imdb.com/ 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Representation & Query Language Such graphs can be represented in relational or ER models. XML with Xlinks, or in the form of RDF triples. They can queried using languages like: SQL, XQuery, or SPARQL. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Closet Relationship Given a set of entities (nodes), find their closet relationships (edges or paths) that connect the entities in the strongest possible way. For example, when asking “How are Germany’s chancellor Angela Merkel, the mathematician Richard Courant, Turing-Award winner Jim Gray, and the Dalai Lama related?”. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Problem Definition To compute the k lowest-cost Steiner trees: Given a graph, with a set of nodes V and a set of edges E, let denote a non- negative weight function. For a given node set, the task is to find the top-k minimum- cost subtrees of G that contain all query nodes of, where the cost of a subtree T with nodes V(T) and edges E(T) is defined as 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Today… Present the STAR algorithm Prove that STAR has a worst approximation ratio of O(log(n)) Analyze the complexity of the algorithm Generalize STAR to compute the top-k relation trees Compare STAR with other algorithms & Evaluations Conclusions 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Related Work Relationship queries – queries which ask for relationships between two or more entities For example, keyword proximity search over relational databases, graph search over ER, RDF and other KB, entity relationship queries on the Web, etc 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Another Definition & Approach Formally, the problem of determining the closet interconnections between two, three or more nodes in a graph ST Problem. The Steiner tree problem can be stated as follows: Given a weighted graph G=(V,E) and a set of nodes, called terminals. Find a tree in G of minimal weight such that it contains all the terminals. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Quality of Approximation The quality of an approximation algorithm is measured by the approximation ratio. That is, the ratio between the weight of the tree output by the algorithm and the optimal Steiner tree. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

GST Group Steiner tree is generalization to the Steiner tree problem Given a weighted graph G=(V,E) and a set of groups where each contains nodes for V, find a tree in G of minimal weight such that it contains at least one node from each group. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Existing Approaches Can be categorized according to their strategies: 1)Distance Network Heuristic (DNH) 2)Span and Cleanup 3)Dynamic Programming 4)Local Search 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

DNH This heuristic, builds a complete graph on the terminals. The edge weights reflect the shortest distance between two terminals in the underlying graph. By a MST heuristic we can construct 2(1-1/n) approximation to the optimal Steiner tree. Moderate size, can fit in the main memory. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Span and Cleanup Heuristic The aim is to build MST starting from an arbitrary terminal and spanning the tree stepwise until it covers all terminals. Redundant nodes are deleted in the cleanup phase. Approximation ratio is O(n) 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Dynamic Programming & DPBF Computing optimal results for all subsets of terminals. Then, the optimal result is computed for all the terminals. This heuristic is modified to a faster method, DPBF, for the optimal solution in the GST case. Both methods are applicable to graphs of moderate size. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Local Search Used in the realm of the Euclidean ST problem and in the parallel ST computation. In the first phase an interconnecting tree is built based on the distance network heuristic. In the second phase the current tree is iteratively improved by considering different nodes in the underlying graph that may improve the cost of the current tree. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Algorithms for top-k ST Computation Top-k ST computation has been previously studied in the context of keyword search on relational databases. There are mainly two approaches: – BANKS (two versions, BANKS 1 & 2) – BLINKS 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

BANKS 1 BANKS 1 addresses the GST problem on directed graphs. It emulates the DNH heuristic by running single source shortest paths iterators from each node in each of the, where is the set of nodes which contain the keyword. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

BANKS 1 The iterators are expanded in a best-first strategy and follow the edges backwards. Solution Detection: As soon as the iterators meet, a result is produced. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

BANKS 2 BANKS 1 technique is improved in BANKS 2 by: – Reducing the number of iterators. – Allowing forward expansion on edges in addition to backwards expansion. – Using a heuristic of spreading activation which prioritizes nodes with low degrees, and edges with low weights during the expansion of iterators. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

BLINKS Makes use of the backward search strategy of BANKS, but based on cost-based expansion. Two kinds of indexes are built to speed up the search: – A keyword node index is built which stores, for each keyword w, a list of nodes that can reach w along with the distance of each node from w. – A node-keyword index is built which stores, for each node, the set of keywords reachable from it and its distance to each keyword. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

BLINKS: Representation Since the proposed indexes can be too large to store and too expensive to compute, the graph is portioned into blocks. The block are formed by partitioning the graph using node separators (also called portals). A high level keyword-block index is built, and more detailed indexes are built at the block level. Multiple cursors are used to perform the backward search within blocks. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

BLINKS: Search technique Whenever a portal of a block is reached, new cursors are created to explore the remaining blocks connecting to this portal node. Instead of trees, BLINKS return pairs, where is the root of the result tree and is a set of nodes containing the query keyword. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

BLINKS: Disadvantages It is difficult to reconstruct the result tree. Need to have the graph in memory to partition it and to construct the indexes. The performance of BLINKS is dependent on the number of portals and the strategy for choosing them. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Outline Introduction Related Work The STAR Algorithm Approximation Guarantee Approximate Top-K Interconnections Time Complexity Evaluation Conclusion 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

The problem Given G=(V,E) (where is V=nodes and E=edges) and a non negative function.For any subgraph of we denote the set of nodes of by, and the set of edges by. Furthermore we extend the weight function on by Give a set, we are interested in finding a subgraph of that contains all nodes from, such that the weight if is minimal among all possible subgraphs of that contain all nodes from 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Representation Many real world graph come with semantic annotations such as node labels, representing entities, and edge labels representing relations. These graphs, may have taxonomic substructures indicated by the labels of the corresponding edges. STAR can exploit such taxonomic backbones, when available, to efficiently find an approximation solution to the problem. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Phase 1 & 2 STAR runs in two phases:  First Phase – tries to quickly build a first tree that interconnect all nodes from.  Second Phase – it aims to iteratively improve the current tree by scanning and pruning its neighborhood. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

First Phase In order to build first interconnecting tree, STAR relies on a similar strategy as BANKS 1. But, instead of running single source shortest path iterators from each node, STAR runs simple BFS iterators from each terminal. The iterators are called in a round-robin manner. As soon as the iterators meet, a result is constructed. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

First Phase Unlike BANKS 1, STAR may exploit taxonomic information (when available) to quickly build a first tree. Taxonomic edges are edges labeled by taxonomic relations such as type or subClassOf. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

First Phase For example, suppose that ={Max Planck, Arnold Schwarzenegger, Germany} In the first phase STAR would construct a tree like this: 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Second Phase In this phase, STAR aims at improving the current tree iteratively by replacing certain paths in the tree by new paths of lower weight cost. In the next few slides we will define which paths can be replaced. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Second Phase: Fixed Nodes and Loose Paths Let T be a tree interconnecting all nodes of. We denote the degree of a node v in T by deg(v). A node is called a terminal node, all other nodes of T called Steiner node. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Second Phase: Fixed Nodes and Loose Paths Definition 1 (fixed node): A node in T is a fixed node if it is either a terminal node or a Steiner node that has degree deg(v)≥3. Definition 2 (loose path): A path p in T is a loose path if it has minimal length with respect to the following property: its end nodes are fixed nodes. A loose path is a path can be replaced in T during the improvement process. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Second Phase: Observations For a tree T, we denote the set of loose paths of a tree T by LP(T). Removing a loose path lp from T splits T into two subtrees T1 and T2. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Second Phase: Example 1 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Second Phase: Lemma 1 A tree T with terminal set,, has at least and at most loose paths. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Second Phase: Finding an Approximate ST STAR keeps iteratively improving the current tree T. In each iteration the algorithm removes a lose path lp from the current tree T. In each iteration T is decomposed into two components T1 and T2. The new tree T is obtained by connecting T1 and T2. through a path that is shorter than lp. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Example continued 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Problem Reduction Hence, the ST problem is reduced to problem of finding shortest paths between subsets of nodes. Heuristically, in each iteration we remove the loose path with maximum weight in T. A high-level overview is given in Algorithm 1. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Algorithm 1 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Example Suppose that ={Max Planck, Arnold Schwarzenegger, Germany} In the first phase, STAR will return: 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Example In the first iteration the algorithm would remove the loose path that connects the fixed node labeled with Germany to the fixed node labeled with person. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Example In the second iteration the connecting the fixed node labeled with Arnold Schwarzenegger to the fixed node labeled with physicist is removed. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Algorithm 2 The method Replace(lp,T) in algorithm 1 removes the loose path lp from T. This removal splits T into subtrees T1 and T2. Then the shortest path in G that connects T1 and T2 into new tree T’ of lower weight. For the purpose, we need a method that finds the shortest path between two given nodes. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Algorithm 2 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Approximation Guarantee The following section, entails that the approximation ratio for the cost of the final tree returned by STAR is independent of the tree constructed in the first phase. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Approximation Guarantee Lemma 2: Let be the set of loose paths in. For any circular ordering of the terminals in, there is a mapping such that: 1. is defined for all loose path in. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Approximation Guarantee 2.For each loose path P with end points u and v, let T1 and T2 be the two trees obtained by removing from all nodes in P (and their edges), except u and v, then, for some i=1,..,N and one of the nodes belongs to T1, while other one belongs to T2. 3.For each pair of terminals there are at most loose paths mapped to 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Approximation Guarantee Theorem 1: The STAR algorithm is a approximation algorithm for the ST problem. Proof: Consider a walk on T0 that uses each edge exactly twice and that visits all nodes in T0. Such walk gives a circular ordering of the terminals, ordered according to their first occurrence in such a walk. We have that (1) 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Proof (continued) Using Lemma 2, we define a mapping with respect to the circular ordering. From property 2 of the mapping and from the termination condition of the STAR algorithm, it follows that for any loose path P=uv in - Where is the distance, in the optimum solution, between the two entries of (2) 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Proof (continued) Finally, we can write: (3) (6) (5) (4) Follows from equation (2) Follows from property 3 of the Mapping u Follows from equation (1) 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Time Complexity The algorithm as it has been presented might have exponential time. In fact, the cost of the tree might decrease at each step by an infinitesimally small amount. This can be solved using a small “trick”. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Improvement-Guarantee Rule Let P be a loose path, and let P’ be the path selected by the algorithm to replace P. Replace P iff The algorithm is then iterated until no loose path can be improved. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Ratio Calculation Let be the maximum and the minimum cost of the edges in the input graph. We will show in Theorem 2 that the STAR algorithm with the improvement-guarantee rule is a pseudo polynomial algorithm. Namely, its running time is polynomial if the ratio is polynomial in the size of the input. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Lemma 3 Let m,n,N denote the number of vertices, edges, and terminals of the input graph respectively. Given, the STAR algorithm with the improvement-guarantee rule is guaranteed to terminate in steps. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Lemma 3: Proof Proof: Let be the initial tree. So Let P be a loose path and let P’ be the path selected by the algorithm to replace P. By the improvement-guarantee rule, it follows that Hence, the cost of the tree decreases at each step by at least. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Lemma 3: Proof (continued) This gives a bound on the number of steps k, as follows: The next theorem shows a trade-off between the approximation guarantee of the STAR algorithm and its running time. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Theorem 2 Given, the STAR algorithm with the improvement-guarantee rule is a approximation algorithm for the ST problem. It running time is 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Theorem 2: Proof Proof: the time-complexity bound follows from Lemma 3 and from the fact that at each step the STAR algorithm might invoke Dijkstra’s algorithm at most (2N-3) times (one for each path according to Lemma 1). To prove the approximation ratio, it suffices to replace Equation 2 in Theorem 1 with 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Theorem 2: Proof (continued) And change the remaining equations accordingly. We include all steps for completeness. We have that: 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Approximate Top-K Interconnections In order to generalize the STAR algorithm, we start from the final tree T returned by the original STAR algorithm, which is stored in priority queue (Algorithm 3). While the size of the priority queue is smaller than k, we keep on generating new trees from an artificial relaxation of the loose path weights of the current tree. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Algorithm 3: getTopK 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Algorithm 4: relax 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Evaluation Datasets: subsets from DBLP and IMDB Queries: three queries, 3,5 and 7 terminals. Top-1 tree comparison for 5 algorithms, STAR, DNH, DPBF, BANKS 1 and BANKS 2. Top-k tree comparison for 4 algorithms STAR, BANKS 1, BANKS 2 and BLINKS. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Evaluation 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Conclusion We have introduced the problem of efficiently answering relationship queries over entity- relation-style data graphs. STAR algorithm can exploit taxonomic structures that are inherent in many KB graphs for fast computation of an initial seed solution. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Conclusion STAR main power for efficiency and result quality comes from iteratively improving the seed tree by a very fast shortest-path algorithm for subtrees defined by the notion of loose path. 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

Thank You For Listening Questions? 28.12.2010 Presented By: Moamen Khiet The Hebrew University Of Jerusalem Seminar on Databases & The Internet

STAR: Steiner-Tree Approximation in Relationship Graphs Presented By: Moamen Khet The Hebrew University Of Jerusalem Seminar on Databases & The Internet.

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STAR: Steiner-Tree Approximation in Relationship Graphs Presented By: Moamen Khet The Hebrew University Of Jerusalem Seminar on Databases & The Internet.

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Presentation on theme: "STAR: Steiner-Tree Approximation in Relationship Graphs Presented By: Moamen Khet The Hebrew University Of Jerusalem Seminar on Databases & The Internet."— Presentation transcript:

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