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S HORTEST P ATH T REE C OMPUTATION IN D YNAMIC G RAPHS Viswanath Gunturi ( ) Bala Subrahmanyam Kambala ( )

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A PPLICATION D OMAIN Transportation Networks:

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S AMPLE D ATASET Sample dataset showing the dynamic nature of Twin Cities road network. Data source: courtesy Navteq Application source: courtesy KwangSoo Yang, Spatial Group, University of Minnesota

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P ROBLEM : D YNAMIC S HORTEST P ATH (DSP) Input: A graph G = (V, E), where V is the set of vertices and E is the set of edges. Each e in E is associated with a positive weight. A source node s and shortest path tree, T s, rooted at node s. A set of changed edges and their new weights. Output: A spanning tree T s ’ rooted a node s. Objective: Computational efficiency. Constraint: The tree T s ’ contains shortest path from node s to all the node v in V.

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Output Input P ROBLEM I LLUSTRATION

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C ONTRIBUTIONS The following three new algorithms were proposed for the DSP problem [9]: DynDijkstra: A generalized version of Dijkstra’s which allows multiple edge weight updates. MBallStringInc: Modified version of existing BallString algorithm [1,2] which ensures correctness. MFP: Modified version of existing DynamicsSWSF-FP algorithm [3] to improve computational efficiency. Extensive experimental analysis for the proposed algorithms on both real and synthetic datasets. Comment on contributions Proposed methods are general. Related work is limited by the some key assumptions such as planarity [4,5], un weighted edges etc [6]. Provides extensive experimental analysis which highlights the superior performance of proposed methods.

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Output Step 2 Step 1 The edge weight increases and decreases are handled separately. Case: Increase only K EY C ONCEPTS : D YN D IJKSTRA Decrease case: Cannot predict affected nodes. All descendants of a affected head node in graph are checked.

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K EY C ONCEPTS :MB ALL S TRING Similar to DynDijkstra algorithm. The priority queue is sorted on the difference between new distance and old distance. This helps in closing an entire branch instead of individual node. Consequently, MBallString has fewer iterations than DynDijkstra.

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V ALIDATION M ETHODOLOGY Correctness proof of DynDijkstra and MBallStringInc algorithm. Extensive experimental analysis of the proposed approaches using both real and synthetic datasets. Real Dataset: Connecticut road system extracted from the US Census Bureau Tiger Line files. Five different sizes: 1K, 2K, 4K, 8K, and 15K number of nodes. A random set of edges is chosen and their weights are modified. Synthetic networks are generated using [7]. The generator assigns edges to vertices such that outgoing degrees of vertices follow quasi-power-law distribution.

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E XPERIMENTAL S ETUP (1/2) Experimental Goals: Evaluate the performance/scalability of proposed algorithms by varying different parameters. Determine the best algorithm for different graph sizes and # changes. Experimental Parameters: Graph size: # vertices Percentage of changed edges (pce) Percentage of changed weight (pcw): % increase or decrease in weight Percentage of increased edges (pie): The ratio between the number of the increased edges and the number of the decreased edges for mixed edge changes. Candidate Algorithms: DynDikstra MBallStringInc MFP Dijkstra

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E XPERIMENTAL S ETUP (2/2) Metrics of Evaluation: Metric 1: CPU runtime for each solution. Metric 2: Total number of operations (given in Table 1) performed by algorithm. Table 1: Unit Operations

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E XPERIMENTAL R ESULTS The parameter pcw has very little effect. Dynamic algorithms better than static Dijkstra when pce is below a certain threshold. This threshold changes with graph sizes. MBallString and DynDijkstra have comparable performance for weight increases. DynDijkstra performs best for weight decreases. MBSDD, a combination of MBallString increase and DynDijkstra decrease, performs best for road networks. DynDijkstra works best for random graphs.

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S TRENGTHS AND W EAKNESSES OF V ALIDATION M ETHODOLOGY Strengths: Both real and synthetic datasets were used for analysis. The experimental parameters capture the experimental goals. Multidimensional variation help to understand the effect of the different parameters. Weaknesses: The set of edges whose weight would be modified (an input to problem) is chosen randomly. May not capture some real world semantics of dynamic networks, such as spatial auto-correlation. For example, in case of road networks, during morning rush hours there would be traffic congestion on roads leading to downtown. Reason for the choice: Comparison with related work. Unavailability of model which can capture dynamic nature of networks for a particular application domain.

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Assumptions: A notion of travel time is not associated with the edges. The paths computed are not Lagrangian in nature, i.e., the cost of an edge should be considered at the time when object/flow arrives at the tail node. Repercussions of Assumptions: Lack of travel time notion → Lagrangian models cannot be used to represent the total cost of path. Non Lagrangian assumption → unsuitable for application domains where the flow is modeled, e.g. transportation networks [8]. Furthermore, the proposed methods are suitable only for shortest paths trees. This may not desirable for real world traveler route planning systems A SSUMPTIONS AND C ONSEQUENCES

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P OSSIBLE R EVISIONS What to preserve? The idea of performing incremental updates to generate new tree. More specifically, the design decisions of MBallString algorithm. What to revise? Explore similar ideas in light of domain specific concepts such as spatial autocorrelation etc. Justification This is because, different kinds of networks have different properties, for e.g. spatial networks may exhibit spatial auto correlation. Different kinds of networks (e.g. power networks, social networks, transportation networks etc) may widely differ from each in terms of structure, degree distribution. Thus, it is important to evaluate the algorithms in light of domain specific knowledge.

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R EFERENCES [1] P. Narva´ez, K. Siu, and H. Tzeng, “New Dynamic Algorithms for Shortest Path Tree Computation,” ACM Trans. Networking, vol. 8, no. 6, pp , [2] P. Narva´ez, K. Siu, and H. Tzeng, “New Dynamic SPT Algorithm Based on a Ball-and-String Model,” ACM Trans. Networking, vol. 9, no. 6, pp , [3] G. Ramalingam and T.W. Reps, “An Incremental Algorithm for a Generalization of the Shortest-Path Problem,” J. Algorithms,vol. 21, no. 2, pp , [4] J. Fakcharoemphol and S. Rao, “Planar Graphs, Negative Weight Edges, Shortest Paths, and Near Linear Time,” Proc. 42nd IEEE Ann. Symp. Foundations of Computer Science (FOCS ’01), pp , [5] P. Klein, S. Rao, M. Rauch, and S. Subramanian, “Faster Shortest-Path Algorithms for Planar Graphs,” Proc. 26th Ann. ACM Symp. Theory of Computing (STOC ’94), pp , [6] S. Baswana, R. Hariharan, and S. Sen, “Improved Decremental Algorithms for Maintaining Transitive Closure and All-Pairs Shortest Paths,” Proc. 34th Ann. ACM Symp. Theory of Computing (STOC ’02), pp , [7] Java Universal Network/Graph Framework, June [8] George, B., Kim, S., Shekhar, S.: Spatio-temporal network databases and routing algorithms: A summary of results. In: Symposium on Spatial and Temporal Databases (SSTD). pp. 460–477 (2007). [9] Edward P.F. Chan, Yaya Yang, "Shortest Path Tree Computation in Dynamic Graphs," IEEE Transactions on Computers, pp , April, 2009

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