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Given a set P = { p 1 , p 2 , … , p n } of n points in Image 1 and each assigned with one of the given k distinct colors, we study the problem of finding the minimum width color spanning annulus of different shapes. Specifically, we consider the color spanning circular annulus ( C S C A ), axis-parallel square annulus ( C S S A ), axis parallel rectangular annulus ( C S R A ) , and equilateral triangular annulus of fixed orientation ( C S E T A ) . The time complexities of the proposed algorithms for the respective problems are (i) O ( n 3 log ⁡ n ) for C S C A , (ii) O ( n 3 + n 2 k log ⁡ k ) for C S S A , (iii) O ( n 4 ) for C S R A , and (iv) O ( n 2 k ) for C S E T A . [ABSTRACT FROM AUTHOR]

Prune-and-search is an excellent algorithmic paradigm for solving various optimization problems. We provide a general scheme for prune-and-search technique and show how to implement it in space-efficient manner. We consider both the in-place and read-only model which have several advantages compared to the traditional model of computation. Our technique can be applied to a large number of problems which accept prune-and-search. For examples, we study the following problems each of which has tremendous practical usage apart from theoretical implication: • computing the minimum enclosing circle (MEC) of a set of n points in R 2 , and • linear programming problems with two and three variables and n constraints. In the in-place setting, all these problems can be solved in O ( n ) time using O ( 1 ) extra-space. In the read-only setup, the time and extra-space complexities of the proposed algorithms for all these problems are O ( n polylog ( n ) ) and O ( polylog ( n ) ) , respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of designing in-place algorithms for finding the maximum clique in the intersection graphs of axis-parallel rectangles and disks in R 2 . First, we propose an O ( n 2 log n ) time in-place algorithm for finding the maximum clique of the intersection graph of a set of n axis-parallel rectangles of arbitrary sizes. For the intersection graph of fixed height rectangles, the time complexity can be slightly improved to O ( n log n + n K ) , where K is the size of the maximum clique. For disk graphs, we consider two variations of the maximum clique problem, namely geometric clique and graphical clique. The time complexity of our algorithm for finding the largest geometric clique is O ( m log n + n 2 ) where m is the number of edges in the disk graph, and it works for disks of arbitrary radii. For graphical clique, our proposed algorithm works for unit disks (i.e., of same radii) and the worst case time complexity is O ( n 2 + m ( n + K 3 ) ) ; m is the number of edges in the unit disk intersection graph and K is the size of the largest clique in that graph. It uses O ( n 3 ) time in-place computation of maximum matching in a bipartite graph, where the vertices are given in an array, and the existence of an edge between a pair of vertices can be checked by an oracle on demand (from problem specification) in O ( 1 ) time. This problem is of independent interest. All these algorithms need O ( 1 ) work space in addition to the input array. [ABSTRACT FROM AUTHOR]

Algorithms and Discrete Mathematics - celebrating the silver jubilee of IITG Guwahati. [Extracted from the article]

A VLSI chip is fabricated by integrating several rectangular circuit blocks on a 2D silicon floor. The circuit blocks are assumed to be placed isothetically and the netlist attached to each block is given. For wire routing, the terminals belonging to the same net are to be electrically interconnected using conducting paths. A staircase channel is an empty polygonal region on the silicon floor bounded by two monotonically increasing (or decreasing) staircase paths from one corner of the floor to its diagonally opposite corner. The staircase paths are defined by the boundaries of the blocks. In this paper, the problem of determining a monotone staircase channel on the floorplan is considered such that the number of distinct nets whose terminals lie on both sides of the channel, is minimized. Two polynomial-time algorithms are presented based on the network flow paradigm. First, the simple two-terminal net model is considered, i.e., each net is assumed to connect exactly two blocks, for which an O(n×k) time algorithm is proposed, where n and k are respectively the number of blocks and nets on the floor. Next, the algorithm is extended to the more realistic case of multi-terminal net problem. The time complexity of the latter algorithm is O((n+k)×T), where T is the total number of terminals attached to all nets in the floorplan. Solutions to these problems are useful in modeling the repeater block placement that arises in interconnect-driven floorplanning for deep-submicron VLSI physical design. It is also an important problem in context to the classical global routing, where channels are used as routing space on silicon. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for identifying a maximal empty-staircase-polygon (MESP) of largest area, among a set of n points on a rectangular floor. A staircase polygon is an isothetic polygon bounded by two monotonically rising (falling) staircases. A monotonically rising staircase is a sequence of alternatingly horizontal and vertical line segments from the bottom-left corner of the floor to its top-right corner such that for every pair of points α=(xα,yα) and β=(xβ,yβ) on the staircase, xα⩽xβ implies yα⩽yβ. A monotonically falling staircase can similarly be defined from the bottom-right corner of the floor to its top-left corner. An empty staircase polygon is a MESP if it is not contained in another larger empty staircase polygon. The problem of recognizing the largest MESP is formulated using permutation graph, and a simple O(n3) time algorithm is proposed. Next, based on certain novel geometric properties of the problem, an improved algorithm is developed that identifies the largest MESP in O(n2) time and space. The algorithm can be easily tailored for identifying the widest MESP in a similar environment. The general problem of locating the largest area/width MESP among a set of isothetic polygonal obstacles, can be solved easily. These geometric optimization problems have several applications to VLSI layout design, robot motion planning, to name a few. [Copyright &y& Elsevier]

We present an algorithm for finding k nearest neighbors of a given query line among a set of n points distributed arbitrarily on a two-dimensional plane. Our algorithm requires O(n2) time and O(n2/log n) space to preprocess the given set of points, and it answers the query for a given line in O(k+log n) time, where k may also be an input at the query time. Almost a similar technique works for finding k farthest neighbors of a query line, keeping the time and space complexities invariant. We also show that if k is known at the time of preprocessing, the time and space complexities for the preprocessing can be reduced keeping the query times unchanged. [Copyright &y& Elsevier]

This work generalizes the classical problem of finding the largest empty rectangle among obstacles in 2D. Given a set P of n points, here a maximal empty rectangle (MER) is defined as a rectangle of arbitrary orientation such that each of its four boundaries contain at least one member of P and the interior of the rectangle is empty. We propose a very simple algorithm based on standard data structure to locate a MER of largest area in the plane. The worst-case time complexity of our algorithm is O(n3). Though the worst-case space complexity is O(n2), it reserves O(nlogn) space on an average to maintain the required data structure during the execution of the algorithm. [Copyright &y& Elsevier]

Classical separability problem involving multi-color point sets is an important area of study in computational geometry. In this paper, we study different separability problems for bichromatic point set P = P r ∪ P b in R 2 and R 3 , where P r and P b represent a set of n red points and a set of m blue points respectively, and the objective is to compute a monochromatic object of a desired type and of maximum size. We propose in-place algorithms for computing (i) an arbitrarily oriented monochromatic rectangle of maximum size in R 2 , and (ii) an axis-parallel monochromatic cuboid of maximum size in R 3. The time complexities of the algorithms for problems (i) and (ii) are O (m (m + n) (m n + m log m + n log n)) and O (m 3 n + m 2 n log n) , respectively. As a prerequisite, we propose an in-place construction of the classic data structure the k-d tree , that was originally invented by J. L. Bentley in 1975. Our in-place variant of the k -d tree for a set of n points in R k supports orthogonal range counting query using O (1) extra workspace, and with O (n 1 − 1 ∕ k) query time complexity. The construction time of this data structure is O (n log n). Both the construction and query algorithms are non-recursive in nature that do not need O (log n) size recursion stack compared to the previously known construction algorithm for in-place k -d tree and query in it. We believe that this result is of independent interest. We also propose an algorithm for the problem of computing an arbitrarily oriented rectangle of maximum weight among a point set P = P r ∪ P b , where each point in P b (resp. P r) is associated with a negative (resp. positive) real-valued weight that runs in O (m 2 (n + m) log (n + m)) time using O (n) extra space. [ABSTRACT FROM AUTHOR]

In this paper, we study some fundamental facility location problems from the space-efficient perspective. We show that 1-center problem in Euclidean space and in tree networks can be efficiently solved in constant-workspace model. We use a virtual pairing tree during the pruning stage that allows the maintenance of pruned elements. We also show that a feasible region during the prune-and-search process can always be maintained using O (1) space. These results realize the solutions to the problems in constant work-space model: linear programming in 2-d, 3-d, 2-center in trees, and largest disk inside a convex polygon. [ABSTRACT FROM AUTHOR]

In this paper, we study the Shortest Color Spanning t - Intervals problem, and related generalizations, namely Smallest Color Spanning t - Squares and Smallest Color Spanning t - Circles. The generic setting is the following: we are given n points in the plane (or on a line), each colored with one of k colors. For each color i we also have a demand s i. Given a budget t , we are required to find at most t objects (for example, intervals, squares, circles, etc.) that cover at least s i points of color i. Typically, the goal is to minimize the maximum perimeter or area. We provide exact algorithms for these problems for the cases of intervals, circles and squares, generalizing several known results. In the case of intervals, we provide a comprehensive understanding of the complexity landscape of the problem after taking several natural parameters into account. Given that the problem turns out to be W 1 -hard parameterized by the standard parameters, we introduce a new parameter, namely sparsity, and prove new hardness and tractability results in this context. For squares and circles, we use existing algorithms of one smallest color spanning object in order to design algorithms for getting t identical objects of minimum size whose union spans all the colors. [ABSTRACT FROM AUTHOR]

Highlights • Algorithms for covering and hitting a given set of line segments by two congruent squares of minimum size is studied. • The algorithm produces optimum solution by sequentially reading the input array (read only) twice. • The extra space complexity of the algorithm is O (1). • A restricted version of covering is also studied where each object is to be completely covered by at least one square. • This algorithm gives 2 approximation result to cover/hit the segments with two congruent disks of minimum size. Abstract This paper discusses the problem of covering and hitting a set of line segments L in R 2 by a pair of axis-parallel congruent squares of minimum size. We also discuss the restricted version of covering, where each line segment in L is to be covered completely by at least one square. The proposed algorithms assume that the input segments are given in a read-only array. For each of these problems (i.e. covering , hitting and restricted covering problems), our proposed algorithm reports the optimum result by executing only two passes of reading the input array sequentially. All these algorithms need only O (1) extra space. The solution of these problems also give a 2 approximation for covering and hitting those line segments L by two congruent disks of minimum radius with same computational complexity. [ABSTRACT FROM AUTHOR]

In the acrophobic guard watchtower problem for a polyhedral terrain, a square axis-aligned platform is placed on the top of a tower whose bottom end-point lies on the surface of the terrain. As in the standard watchtower problem, the objective is to minimize the height (i.e., the length) of the watchtower such that every point on the surface of the terrain is weakly visible from the platform placed on the top of the tower. In this paper, we show that in R 2 the problem can be solved in O (n) time, and in R 3 it takes O (n log ⁡ n) time, where n is the total number of vertices of the terrain. [ABSTRACT FROM AUTHOR]

Let P be a set of n points in general position in the plane which is partitioned into color classes. The set P is said to be color-balanced if the number of points of each color is at most [n/2] . Given a color-balanced point set P, a balanced cut is a line which partitions P into two color-balanced point sets, each of size at most 2n/3+1 . A colored matching of P is a perfect matching in which every edge connects two points of distinct colors by a straight line segment. A plane colored matching is a colored matching which is non-crossing. In this paper, we present an algorithm which computes a balanced cut for P in linear time. Consequently, we present an algorithm which computes a plane colored matching of P optimally in Θ(nlog n) time. [ABSTRACT FROM AUTHOR]

Given a set of n colored points in IR2 with a total of m (3 ≤ m ≤ n) colors, the problem of identifying the smallest color-spanning object of some predefined shape is studied in this paper. We shall consider two different shapes: (i) corridor and (ii) rectangle of arbitrary orientation. Our proposed algorithm for identifying the smallest color-spanning corridor is simple and runs in O(n2log n) time using O(n) space. A dynamic version of the problem is also studied, where new points may be added, and the narrowest color-spanning corridor at any instance can be reported in O(mn(α(n))2log m) time. Our algorithm for identifying the smallest color-spanning rectangle of arbitrary orientation runs in O(n3log m) time and O(n) space. [ABSTRACT FROM AUTHOR]

Abstract: Given a polyhedral terrain with n vertices, the shortest monotone descent path problem deals with finding the shortest path between a pair of points, called source (s) and destination (t) such that the path is constrained to lie on the surface of the terrain, and for every pair of points and on the path, if then , where denotes the distance of p from s along the aforesaid path. This is posed as an open problem by Berg and Kreveld [M. de Berg, M. van Kreveld, Trekking in the Alps without freezing or getting tired, Algorithmica 18 (1997) 306–323]. We show that for some restricted classes of polyhedral terrain, the optimal path can be identified in polynomial time. [Copyright &y& Elsevier]

In the unbounded version of the range assignment problem for all-to-all communication in 1D, a set of n radio-stations are placed arbitrarily on a line; the objective is to assign ranges to these radio-stations such that each of them can communicate with the others (using at most n - 1 hops) and the total power consumption is minimum. A simple incremental algorithm for this problem is proposed which produces optimum solution in O(n3) time and O(n2) space. This is an improvement in the running time by a factor of n over the best known existing algorithm for the same problem. [ABSTRACT FROM AUTHOR]

Abstract: In the context of mobile communication, an efficient algorithm for the base-station placement problem is developed in this paper. The objective is to place a given number of base-stations in a given convex region, and to assign range to each of them such that every point in the region is covered by at least one base-station, and the maximum range assigned is minimized. It is basically covering a region by a given number of equal radius circles where the objective is to minimize the radius. We develop an efficient algorithm for this problem using Voronoi diagram which works for covering a convex region of arbitrary shape. Existing results for this problem are available when the region is a square [K.J. Nurmela, P.R.J. Ostergard, Covering a square with up to 30 equal circles, Research Report HUT-TCS-A62, Laboratory for Theoretical Computer Science, Helsinky University of Technology, 2000] and an equilateral triangle [K.J. Nurmela, Conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles, Exp. Math. 9 (2) (2000)]. The minimum radius obtained by our method favorably compares with the results presented in [K.J. Nurmela, P.R.J. Ostergard, Covering a square with up to 30 equal circles, Research Report HUT-TCS-A62, Laboratory for Theoretical Computer Science, Helsinky University of Technology, 2000; K.J. Nurmela, Conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles, Exp. Math. 9 (2) (2000)]. But the execution time of our algorithm is a fraction of a second, whereas the existing methods may even take about two weeks’ time for a reasonable value of the number of circles () as reported in [K.J. Nurmela, P.R.J. Ostergard, Covering a square with up to 30 equal circles, Research Report HUT-TCS-A62, Laboratory for Theoretical Computer Science, Helsinky University of Technology, 2000; K.J. Nurmela, Conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles, Exp. Math. 9 (2) (2000)]. [Copyright &y& Elsevier]

Abstract: This paper presents simple and deterministic algorithms for partial point set pattern matching in 2D. Given a set P of n points, called sample set, and a query set Q of k points (), the problem is to find a matching of Q with a subset of P under rigid motion. The match may be of two types: exact and approximate. If an exact matching exists, then each point in Q coincides with the corresponding point in P under some translation and/or rotation. For an approximate match, some translation and/or rotation may be allowed such that each point in Q lies in a predefined -neighborhood region around some point in P. The proposed algorithm for the exact matching needs space and preprocessing time. The existence of a match for a given query set Q can be checked in time in the worst-case, where is the maximum number of equidistant pairs of point in P. For a set of n points, may be in the worst-case. Some applications of the partial point set pattern matching are then illustrated. Experimental results on random point sets and some fingerprint databases show that, in practice, the computation time is much smaller than the worst-case requirement. The algorithm is then extended for checking the exact match of a set of k line segments in the query set with a k-subset of n line segments in the sample set under rigid motion in time. Next, a simple version of the approximate matching problem is studied where one point of Q exactly matches with a point of P, and each of the other points of Q lie in the -neighborhood of some point of P. The worst-case time and space complexities of the proposed algorithm are and , respectively. The proposed algorithms will find many applications to fingerprint matching, image registration, and object recognition. [Copyright &y& Elsevier]

Abstract: Given a set S of n radio-stations located on a d-dimensional space, a source node s and an integer h , the h-hop broadcast range assignment problem deals with assigning ranges to the members in S so that s can communicate with all other members in S in at most h-hops, and the total power consumption is minimum. The problem is known to be NP-hard for . We propose an time algorithm for the one dimensional version () of the problem. This is an improvement over the existing result on this problem by a factor of h [A.E.F. Clementi et al. The minimum broadcast range assignment problem on linear multi-hop wireless networks, Theoret. Comput. Sci. 299 (2003) 751–761]. [Copyright &y& Elsevier]

We present an efficient algorithm for finding k nearest neighbors of a query line segment among a set of points distributed arbitrarily on a two dimensional plane. Along the way to finding this algorithm, we have obtained an improved triangular range searching technique in 2D. Given a set of n points, we preprocess them to create a data structure using O(n2) time and space, such that given a triangular query region Δ, the number of points inside Δ can be reported in O(logn) time. Finally, this triangular range counting technique is used for finding k nearest neighbors of a query straight line segment in O(log2n+k) time. [Copyright &y& Elsevier]

Given a convex polygon with n vertices, we study the problem of identifying a triangle with its smallest side as large as possible among all the triangles that can be drawn inside the polygon. We show that at least one of the vertices of such a triangle must coincide with a vertex of the polygon. We also propose an O (n 2) time algorithm to compute such a triangle inside the given convex polygon. [ABSTRACT FROM AUTHOR]

In the paper "Linear time algorithm to cover and hit a set of line segments optimally by two axis-parallel squares", Theor. Comput. Sci. 769 (2019) 63–74, the LHIT problem is proposed as follows: For a given set of non-intersecting line segments L = { ℓ 1 , ℓ 2 , ... , ℓ n } in Image 1 , compute two axis-parallel congruent squares S 1 and S 2 of minimum size whose union hits all the line segments in L , and a linear time algorithm was proposed. Later it was observed that the algorithm has a bug. In this corrigendum, we corrected the algorithm. The time complexity of the corrected algorithm is O (n 2). [ABSTRACT FROM AUTHOR]

We study the rectilinear path problem in the presence of disjoint axis parallel rectangular obstacles in the read-only and in-place setup. The input to the problem is a set R of n axis-parallel rectangular obstacles in R 2 . The objective is to answer the following query efficiently. Path-Query ( p , q ) : Given a pair of points p and q , report an axis-parallel path from p to q avoiding the obstacles in R . In the read-only setup, we show that Path-Query ( p , q ) problem can be solved in O ( n 2 s + n log s ) time using O ( s ) extra space. We also show that the existence of an x -monotone path and reporting it, if it exists, can be done with the same asymptotic time complexity. If the objective is to test the existence of an x y -monotone path between the given pair of points p and q avoiding the obstacles, and report it if exists, then our proposed algorithm needs O ( n 2 s + n log s + M s log n ) time with O ( s ) extra space, where M s is the time complexity for computing the median of n elements in the read-only setup using O ( s ) extra space. Finally, we show that when the obstacles are unit squares instead of rectangles of arbitrary size, then there always exists a path of O ( n ) links between a pair of query points, and the path can be reported in O ( n n ) time using O ( 1 ) extra work-space. It is also shown that there is an instance where the minimum number of links in a path between a pair of specified points is O ( n ) . The objective of the Path-Query ( p , q ) in the in-place setup is to preprocess the input rectangles in a data structure in the input array itself such that for any pair of query points p and q , a rectilinear path can be reported efficiently. Here we propose an algorithm with O ( n log n ) preprocessing time and O ( n 3 / 4 + χ ) query time, where χ is the number of links (bends) in the path. Both the preprocessing and query answering need O ( 1 ) extra space. [ABSTRACT FROM AUTHOR]