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**11 - 16**of**16**### Undiscretized Dynamic Programming: Faster Algorithms for Facility Location and Related Problems on Trees

"... In the Uncapacitated Facility Location (UFL) problem, there is a xed cost for opening a facility, and some distance matrix d that determines the cost of distributing commodities from any facility i to any consumer j. The problem is NP-hard in general and when d consists of a distance metric in a gra ..."

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In the Uncapacitated Facility Location (UFL) problem, there is a xed cost for opening a facility, and some distance matrix d that determines the cost of distributing commodities from any facility i to any consumer j. The problem is NP-hard in general and when d consists of a distance metric in a graph [7, 12]. However, for the case where the commodity transportation costs are given by path lengths in a tree, an O(n 2) dynamic programming algorithm was given by [4,7]. We improve this dynamic programming algorithm by using the geometry of piecewise linear functions and fast merging of the binary search trees used to store these functions. We achieve the complexity bound of O(n log n) fortheTree Location Problem and some related problems. Our approach gives a general method for solving tree dynamic programming problems. 1

### Undiscretized Dynamic Programming: Faster Algorithms for Facility Location and Related Problems on Trees

, 2002

"... In the Uncapacitated Facility Location (UFL) problem, there is a fixed cost for opening a facility, and some distance matrix d that determines the cost of distributing commodities from any facility i to any consumer j. The problem is NP-hard in general and when d consists of a distance metric in a g ..."

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In the Uncapacitated Facility Location (UFL) problem, there is a fixed cost for opening a facility, and some distance matrix d that determines the cost of distributing commodities from any facility i to any consumer j. The problem is NP-hard in general and when d consists of a distance metric in a graph [7, 12]. However, for the case where the commodity transportation costs are given by path lengths in a tree, an O(n 2 ) dynamic programming algorithm was given by [4, 7]. We improve this dynamic programming algorithm by using the geometry of piecewise linear functions and fast merging of the binary search trees used to store these functions. We achieve the complexity bound of O(n log n) for the Tree Location Problem and some related problems. Our approach gives a general method for solving dynamic tree programming problems. 1

### Algorithms for Efficient Filtering in Content-Based Multicast

, 2001

"... Content-Based Multicast is a type of multicast where the source sends a set of different classes of information and not all the subscribers in the multicast group need all the information. Use of filtering publish-subscribe agents on the intermediate nodes was suggested [5] to filter out the unn ..."

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Content-Based Multicast is a type of multicast where the source sends a set of different classes of information and not all the subscribers in the multicast group need all the information. Use of filtering publish-subscribe agents on the intermediate nodes was suggested [5] to filter out the unnecessary information on the multicast tree. However, filters have their own drawbacks like processing delays and infrastructure cost. Hence, it is desired to place these filters most efficiently. An O(n 2 ) dynamic programming algorithm was proposed to calculate the best locations for filters that would minimize overall delays in the network [6]. We propose an improvement of this algorithm which exploits the geometry of piecewise linear functions and fast merging of sorted lists, represented by height balanced search trees, to achieve O(n log n) time complexity. Also, we show an improvement of this algorithm which runs in O(n log h) time, where h is the height of the multicast tree. This problem is closely related to p-median and uncapacitated facility location over trees. Theoretically, this is an uncapacitated analogue of the p-inmedian problem on trees as defined in [9].

### Correspondence Based Data Structures for Double Ended Priority Queues

"... this paper is to demonstrate the generality of two techniques used in [6] to develop an MDEPQ representation from an MPQ representation - height biased leftist trees. These methods - total correspondence and leaf correspondence - may be used to arrive at efficient DEPQ and MDEPQ data structures from ..."

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this paper is to demonstrate the generality of two techniques used in [6] to develop an MDEPQ representation from an MPQ representation - height biased leftist trees. These methods - total correspondence and leaf correspondence - may be used to arrive at efficient DEPQ and MDEPQ data structures from PQ and MPQ data structures such as the pairing heap [8; 18], Binomial and Fibonacci heaps [9], and Brodal's FMPQ [2] which also provide efficient support for the operation: --Delete(Q,p): delete and return the element located at p We begin, in Section 2, by reviewing a rather straightforward way, dual priority queues, to obtain a (M)DEPQ structure from a (M)PQ structure. This method [2; 6] simply puts each element into both a minPQ and a maxPQ. In Section 3, we describe the total correspondence method and in Section 4, we describe leaf correspondence. Both sections provide examples of PQs and MPQs and the resulting DEPQs and MDEPQs. Section 5 gives complexity results. In Section 6, we provide the result of experiments that compare the performance of the MDEPQs based on height biased leftist tree [7], pairing heaps [8; 18], and FMPQs [2]. For reference purpose, we also provide run times for the splay tree data structure [16]. Although splay trees were not specifically designed to represent DEPQs, it is easy min Heap max Heap Fig. 1. Dual heap structure to use them for this purpose. Note that splay trees do not provide efficient support for the Meld operation

### Strict Fibonacci Heaps

"... Wepresentthefirstpointer-basedheapimplementationwith time bounds matching those of Fibonacci heaps in the worst case. We support make-heap, insert, find-min, meld and decrease-key in worst-case O(1) time, and delete and deletemin in worst-case O(lgn) time, where n is the size of the heap. The data s ..."

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Wepresentthefirstpointer-basedheapimplementationwith time bounds matching those of Fibonacci heaps in the worst case. We support make-heap, insert, find-min, meld and decrease-key in worst-case O(1) time, and delete and deletemin in worst-case O(lgn) time, where n is the size of the heap. The data structure uses linear space. A previous, very complicated, solution achieving the same time bounds in the RAM model made essential use of arrays and extensive use of redundant counter schemes to maintain balance. Our solution uses neither. Our key simplification is to discard the structure of the smaller heap when doing a meld. We use the pigeonhole principle in place of the redundant counter mechanism.

### Queries and Fault Tolerance

"... The focus of this dissertation is on algorithms, in particular data structures that give provably efficient solutions for sequence analysis problems, range queries, and fault tolerant computing. The work presented in this dissertation is divided into three parts. In Part I we consider algorithms for ..."

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The focus of this dissertation is on algorithms, in particular data structures that give provably efficient solutions for sequence analysis problems, range queries, and fault tolerant computing. The work presented in this dissertation is divided into three parts. In Part I we consider algorithms for a range of sequence analysis problems that have risen from applications in pattern matching, bioinformatics, and data mining. On a high level, each problem is defined by a function and some constraints and the job at hand is to locate subsequences that score high with this function and are not invalidated by the constraints. Many variants and similar problems have been proposed leading to several different approaches and algorithms. We consider problems where the function is the sum of the elements in the sequence and the constraints only bound the length of the subsequences considered. We give optimal algorithms for several variants of the problem based on a simple idea and classic algorithms and data structures. In Part II we consider range query data structures. This a category of