Guido Proietti

Image indexing and retrieval based on human perceptual color clustering

We propose a new image retrieval method based on human perceptual clustering of color images. This color clustering produces for each image a small set of representative colors which captures the color properties of the image, and a small set of sizable contiguous regions which captures the spatial/geometrical properties of the image. The proposed method outperforms the traditional histogram and its improved methods not only with its richer image retrieval capabilities which cover a wider spectrum of user requirements, but also with its powerful indexing scheme which is essential to cater for large scale image databases.

Finding the most vital node of a shortest path

In an undirected, 2-node connected graph G=(V,E) with positive real edge lengths, the distance between any two nodes r and s is the length of a shortest path between r and s in G. The removal of a node and its incident edges from G may increase the distance from r to s. A most vital node of a given shortest path from r to s is a node (other than r and s) whose removal from G results in the largest increase of the distance from r to s. In the past, the problem of finding a most vital node of a given shortest path has been studied because of its implications in network management, where it is important to know in advance which component failure will affect network efficiency the most. In this paper, we show that this problem can be solved in O(m+n log n) time and O(m) space, where m and n denote the number of edges and the number of nodes in G.

A faster computation of the most vital edge of a shortest path

Abstract Let P G (r,s) denote a shortest path between two nodes r and s in an undirected graph G=(V,E) such that |V|=n and |E|=m and with a positive real length w(e) associated with any e∈E . In this paper we focus on the problem of finding an edge e ∗ ∈P G (r,s) whose removal is such that the length of P G−e ∗ (r,s) is maximum, where G−e ∗ =(V,E⧹{e ∗ }) . Such an edge is known as the most vital edge of the path P G (r,s) . We will show that this problem can be solved in O (m·α(m,n)) time, where α is the functional inverse of the Ackermann function, thus improving on the previous O (m+n log n) time bound.

Finding the detour-critical edge of a shortest path between two nodes

Abstract Let PG(r, s) denote a shortest path between two nodes r and s in an undirected graph G with nonnegative edge weights. A detour at a node u ϵ PG(r, s) = 〈r,…, u, v,…,s〉 is defined as a shortest path PG − e(u, s) from u to s which does not make use of (u, v). In this paper we focus on the problem of finding an edge e = (u, v) ϵ PG(r, s) whose removal produces a detour at node u such that the length of PG − e(u, s) minus the length of PG(u, s) is maximum. We call such an edge a detour-critical edge. We will show that this problem can be solved in O(m + n log n) time, where n and m denote the number of nodes and edges in the graph, respectively.

Swapping a failing edge of a shortest paths tree by minimizing the average stretch factor

We consider a two-edge connected, undirected graph G=(V,E), with n nodes and m non-negatively real weighted edges, and a single source shortest paths tree (SPT) T of G rooted at an arbitrary node r. If an edge in T is temporarily removed, it makes sense to reconnect the nodes disconnected from the root by adding a single non-tree edge, called a swap edge, instead of rebuilding a new optimal SPT from scratch. In the past, several optimality criteria have been considered to select a best possible swap edge. In this paper we focus on the most prominent one, that is the minimization of the average distance between the root and the disconnected nodes. To this respect, we present an O(mlog2n) time and O(m) space algorithm to find a best swap edge for every edge of T, thus improving for m=o(n2/log2n) the previously known O(n2) time and space complexity algorithm.

I/O complexity for range queries on region data stored using an R-tree

We study the node distribution of an R-tree storing region data, like for instance islands, lakes or human-inhabited areas. We show that real region datasets are packed in minimum bounding rectangles (MBRs) whose area distribution follows the same power law, named REGAL (REGion Area Law), as that for the regions themselves. Moreover these MBRs are packed in their turn into MBRs following the same law, and so on iteratively, up to the root of the R-tree. Based on this observation, we are able to accurately estimate the search effort for range queries, the most prominent spatial operation, using a small number of easy-to-retrieve parameters. Experiments on a variety of real datasets (islands, lakes, human-inhabited areas) show that our estimation is accurate, enjoying a maximum geometric average relative error within 30%.

Finding the Most Vital Node of a Shortest Path

In an undirected, 2-node connected graph G= (V,E) with positive real edge lengths, the distance between anyt wo nodes r and s is the length of a shortest path between r and s in G. The removal of a node and its incident edges from G may increase the distance from r to s. A most vital node of a given shortest path from rto s is a node (other than r and s) whose removal from G results in the largest increase of the distance from r to s. In the past, the problem of finding a most vital node of a given shortest path has been studied because of its implications in network management, where it is important to know in advance which component failure will affect network efficiencythe most. In this paper, we show that this problem can be solved in O(m+ nlog n) time and O(m) space, where mand n denote the number of edges and the number of nodes in G.

Reusing Optimal TSP Solutions for Locally Modified Input Instances

Given an instance of an optimization problem together with an optimal solution, we consider the scenario in which this instance is modified locally. In graph problems, e.g., a singular edge might be removed or added, or an edge weight might be varied, etc. For a problem U and such a local modification operation, let LM-U (local-modification-U) denote the resulting problem. The question is whether it is possible to exploit the additional knowledge of an optimal solution to the original instance or not, i.e., whether LM-U is computationally more tractable than U. Here, we give non-trivial examples both of problems where this is and problems where this is not the case. Our main results are these: 1. The local modification to change the cost of a singular edge turns the traveling salesperson problem (TSP) into a problem LM-TSP which is as hard as TSP itself, i.e., unless P=NP, there is no polynomial-time p(n)-approximation algorithm for LM-TSP for any polynomial p. Moreover, LM-TSP where inputs must satisfy the β triangle inequality (LM-Δ β -TSP) remains NP-hard for all β > 1/2. 2. For LM-Δ-TSP (i.e., metric LM-TSP), an efficient 1.4-approximation algorithm is presented. In other words, the additional information enables us to do better than if we simply used Christofides’ algorithm for the modified input. 3. Similarly, for all 1 < β < 3.34899, we achieve a better approximation ratio for LM-Δ β -TSP than for Δ’-TSP. 4. Metric TSP with deadlines (time windows), if a single deadline or the cost of a single edge is modified, exhibits the same lower bounds on the approximability in these local-modification versions as those currently known for the original problem. instance. A second construction inflates this advantage. Tours which start at time X, different from those that start between times X+g and X +ςg, may spend some extra time to visit a group of vertices which, unless visited early, will cause belated tours to run k times zigzag across a huge distance γ.

Time and space efficient secondary memory representation of quadtrees

Abstract Efficient management of spatial data is becoming more and more important and for very large sets of 2-dimensional data, secondary memory data representations are required. An important class of queries for spatial data are those that extract a subset of the data: they are called window queries (also region or range queries). In this paper we propose and analyze a new data structure, namely the hybrid linear quadtree, for the efficient secondary memory processing of three kinds of window queries, that is the exist, the report and the select query. In particular we prove that for a window of size n × n in a feature space (e.g., an image) of size T × T using the hybrid linear quadtree stored on a B+-tree with bucket size r, the exist and report query can be answered with O(n logr T) accesses to secondary storage, while the select query can be answered with O(n log r T + n 2 r ) accesses to secondary storage. This is an improvement in worst-case I/O time complexity over previous results and shows that multiple non-overlapping features (i.e., coloured images) can be treated with the same I/O complexity as single features (i.e., black and white images). Furthermore, we show that the hybrid linear quadtree has a low space occupancy overhead with respect to the classic linear quadtree.

Bounded-Distance Network Creation Games

A <i>network creation game</i> simulates a decentralized and noncooperative construction of a communication network. Informally, there are <i>n</i> players sitting on the network nodes, which attempt to establish a reciprocal communication by activating, thereby incurring a certain cost, any of their incident links. The goal of each player is to have all the other nodes as close as possible in the resulting network, while buying as few links as possible. According to this intuition, any model of the game must then appropriately address a balance between these two conflicting objectives. Motivated by the fact that a player might have a strong requirement about her centrality in the network, we introduce a new setting in which a player who maintains her (maximum or average) distance to the other nodes within a given bound incurs a cost equal to the number of activated edges; otherwise her cost is unbounded. We study the problem of understanding the structure of pure Nash equilibria of the resulting games, which we call M<scp>ax</scp>BD and S<scp>um</scp>BD, respectively. For both games, we show that when distance bounds associated with players are nonuniform, then equilibria can be arbitrarily bad. On the other hand, for M<scp>ax</scp>BD, we show that when nodes have a uniform bound <i>D</i> ≥ 3 on the maximum distance, then the <i>price of anarchy</i> (PoA) is lower and upper bounded by 2 and <i>O</i>(<i>n</i>^{1/⌊log₃ <i>D</i> ⌋+1}), respectively (i.e., PoA is constant as soon as <i>D</i> is Ω(<i>n</i>^ε), for any ε > 0), while for the interesting case <i>D</i>=2, we are able to prove that the PoA is Ω(&sqrt;<i>n</i>) and <i>O</i>(&sqrt;<i>n</i> log <i>n</i>). For the uniform S<scp>um</scp>BD, we obtain similar (asymptotically) results and moreover show that PoA becomes constant as soon as the bound on the average distance is 2^{<i>ω</i>(&sqrt;log <i>n</i>)}.