Katedra Algorytmów i Modelowania Systemów

Publikacje

  1. 2018

  2. Collaborative Exploration of Trees by Energy-Constrained Mobile Robots

    S. Das , D. Dereniowski , C. Karousatou - THEORY OF COMPUTING SYSTEMS - 2018
    We study the problem of exploration of a tree by mobile agents (robots) that have limited energy. The energy constraint bounds the number of edges that can be traversed by a single agent. We use a team of agents to collectively explore the tree and the objective is to minimize the size of this team. The agents start at a single node, the designated root of the tree and the height of the tree is assumed to be less than the energy...
    • Pełny tekst w serwisie zewnętrznym
  3. Dynamic F-free Coloring of Graphs

    P. Borowiecki , E. Sidorowicz - GRAPHS AND COMBINATORICS - 2018
    A problem of graph F-free coloring consists in partitioning the vertex set of a graph such that none of the resulting sets induces a graph containing a fixed graph F as an induced subgraph. In this paper we consider dynamic F-free coloring in which, similarly as in online coloring, the graph to be colored is not known in advance; it is gradually revealed to the coloring algorithm that has to color each vertex upon request as well...
    • Pełny tekst w serwisie zewnętrznym
  4. On incidence coloring of coloring of complete multipartite and semicubic bipartite graphs

    In the paper, we show that the incidence chromatic number of a complete k-partite graph is at most ∆+2 (i.e., proving the incidence coloring conjecture for these graphs) and it is equal to ∆+1 if and only if the smallest part has only one vertex.
    • Pełny tekst w serwisie zewnętrznym
  5. Scheduling of unit-length jobs with cubic incompatibility graphs on three uniform machines

    H. Furmańczyk , M. Kubale - DISCRETE APPLIED MATHEMATICS - 2018
    We consider the problem of scheduling n identical jobs on 3 uniform machines with speeds s1, s2, and s3 to minimize the schedule length. We assume that jobs are subject to some kind of mutual exclusion constraints, modeled by a cubic incompatibility graph. We how that if the graph is 2-chromatic then the problem can be solved in O(n^2) time. If the graph is 3-chromatic, the problem becomes NP-hard even if s1>s2=s3.
    • Pełny tekst w serwisie zewnętrznym
  6. Tight bounds on the complexity of semi-equitable coloring of cubic and subcubic graphs

    H. Furmańczyk , M. Kubale - DISCRETE APPLIED MATHEMATICS - 2018
    We consider the complexity of semi-equitable k-coloring, k>3, of the vertices of a cubic or subcubic graph G. In particular, we show that, given a n-vertex subcubic graph G, it is NP-complete to obtain a semi-equitable k-coloring of G whose non-equitable color class is of size s if s>n/3, and it is polynomially solvable if s, n/3.
  7. Turán numbers for odd wheels

    The Turán number ex(n,G) is the maximum number of edges in any n-vertex graph that does not contain a subgraph isomorphic to G. A wheel W_n is a graph on n vertices obtained from a C_{n−1} by adding one vertex w and making w adjacent to all vertices of the C_{n−1}. We obtain two exact values for small wheels: ex(n,W_5)=\lfloor n^2/4+n/2\rfloor, ex(n,W_7)=\lfloor n^2/4+n/2+1 \rfloor. Given that ex(n,W_6) is already known, this...
    • Pełny tekst w serwisie zewnętrznym
  8. 2017

  9. Approximation Strategies for Generalized Binary Search in Weighted Trees

    D. Dereniowski , A. Kosowski , P. Uznański , M. Zou - 2017
    We consider the following generalization of the binary search problem. A search strategy is required to locate an unknown target node t in a given tree T. Upon querying a node v of the tree, the strategy receives as a reply an indication of the connected component of T\{v} containing the target t. The cost of querying each node is given by a known non-negative weight function, and the considered objective is to minimize the total...
    • Pełny tekst w serwisie zewnętrznym
  10. Average distance is submultiplicative and subadditive with respect to the strong product of graphs

    We show that the average distance is submultiplicative and subadditive on the set of non-trivial connected graphs with respect to the strong product. We also give an application of the above-mentioned result.
    • Pełny tekst w serwisie zewnętrznym
  11. Collaborative Delivery by Energy-Sharing Low-Power Mobile Robots

    E. Bampas , S. Das , D. Dereniowski , C. Karousatou - 2017
    We study two variants of delivery problems for mobile robots sharing energy. Each mobile robot can store at any given moment at most two units of energy, and whenever two robots are at the same location, they can transfer energy between each other, respecting the maximum capacity. The robots operate in a simple graph and initially each robot has two units of energy. A single edge traversal by an robot reduces its energy by one...
    • Pełny tekst w serwisie zewnętrznym
  12. Collision-free network exploration

    J. Czyzowicz , D. Dereniowski , L. Gąsieniec , R. Klasing , A. Kosowski , D. Pająk - JOURNAL OF COMPUTER AND SYSTEM SCIENCES - 2017
    Mobile agents start at different nodes of an n-node network. The agents synchronously move along the network edges in a collision-free way, i.e., in no round two agents may occupy the same node. An agent has no knowledge of the number and initial positions of other agents. We are looking for the shortest time required to reach a configuration in which each agent has visited all nodes and returned to its starting location. In...
    • Pełny tekst w serwisie zewnętrznym
  13. Comparing Phylogenetic Trees by Matching Nodes Using the Transfer Distance Between Partitions

    D. Bogdanowicz , K. Giaro - JOURNAL OF COMPUTATIONAL BIOLOGY - 2017
    Ability to quantify dissimilarity of different phylogenetic trees describing the relationship between the same group of taxa is required in various types of phylogenetic studies. For example, such metrics are used to assess the quality of phylogeny construction methods, to define optimization criteria in supertree building algorithms, or to find horizontal gene transfer (HGT) events. Among the set of metrics described so far in...
    • Pełny tekst w serwisie zewnętrznym
  14. Complementarity between entanglement-assisted and quantum distributed random access code

    A. Hameedi , D. Saha , P. Mironowicz , M. Pawłowski , M. Bourennane - PHYSICAL REVIEW A - 2017
    Collaborative communication tasks such as random access codes (RACs) employing quantum resources have manifested great potential in enhancing information processing capabilities beyond the classical limitations. The two quantum variants of RACs, namely, quantum random access code (QRAC) and the entanglement-assisted random access code (EARAC), have demonstrated equal prowess for a number of tasks. However, there do exist specific...
    • Pełny tekst w serwisie zewnętrznym
  15. Computational aspects of greedy partitioning of graphs

    In this paper we consider a variant of graph partitioning consisting in partitioning the vertex set of a graph into the minimum number of sets such that each of them induces a graph in hereditary class of graphs P (the problem is also known as P-coloring). We focus on the computational complexity of several problems related to greedy partitioning. In particular, we show that given a graph G and an integer k deciding if the greedy...
    • Pełny tekst w serwisie zewnętrznym
  16. Equitable coloring of corona multiproducts of graphs

    H. Furmańczyk , M. Kubale , V. Mkrtchyan - Discussiones Mathematicae Graph Theory - 2017
    We give some results regarding the equitable chromatic number for l-corona product of two graphs: G and H, where G is an equitably 3- or 4-colorable graph and H is an r-partite graph, a cycle or a complete graph. Our proofs lead to polynomial algorithms for equitable coloring of such graph products provided that there is given an equitable coloring of G.
  17. Gdańska Międzynarodowa Szkoła Letnia na WETI

    M. Kubale - Pismo PG - 2017
    W dniach 5-12 września 2017 roku Katedra Algorytmów i Modelowania Systemów23, przy wydatnej pomocy,WETI, zorganizowała Międzynarodową Szkołę Letnią poświęconą algorytmom dla problemów optymalizacji dyskretnej.
  18. Interval incidence coloring of subcubic graphs

    In this paper we study the problem of interval incidence coloring of subcubic graphs. In [14] the authors proved that the interval incidence 4-coloring problem is polynomially solvable and the interval incidence 5-coloring problem is N P-complete, and they asked if χii(G) ≤ 2∆(G) holds for an arbitrary graph G. In this paper, we prove that an interval incidence 6-coloring always exists for any subcubic graph G with ∆(G) = 3.
    • Pełny tekst w serwisie zewnętrznym
  19. Monitoring of the Process of System Information Broadcasting in Time

    One of the problems of quantum physics is how a measurement turns quantum, noncopyable data, towards copyable classical knowledge. We use the quantum state discrimination in a central system model to show how its evolution leads to the broadcasting of the information, and how orthogonalization and decoherence factors allow us to monitor the distance of the state in question to the one perfectly broadcasting information, in any...
    • Pełny tekst w serwisie zewnętrznym
  20. No-Wait & No-Idle Open Shop Minimum Makespan Scheduling with Bioperational Jobs

    In the open shop scheduling with bioperational jobs each job consists of two unit operations with a delay between the end of the first operation and the beginning of the second one. No-wait requirement enforces that the delay between operations is equal to 0. No-idle means that there is no idle time on any machine. We model this problem by the interval incidentor (1, 1)-coloring (IIR(1, 1)-coloring) of a graph with the minimum...
    • Pełny tekst w serwisie zewnętrznym
  21. On Computational Aspects of Greedy Partitioning of Graphs

    P. Borowiecki - 2017
    In this paper we consider a problem of graph P-coloring consisting in partitioning the vertex set of a graph such that each of the resulting sets induces a graph in a given additive, hereditary class of graphs P. We focus on partitions generated by the greedy algorithm. In particular, we show that given a graph G and an integer k deciding if the greedy algorithm outputs a P-coloring with a least k colors is NP-complete for an infinite...
    • Pełny tekst w serwisie zewnętrznym
  22. Realizacja zadań w grafie przez grupę mobilnych jednostek

    D. Osula - 2017
    Grupa mobilnych jednostek, nazywanych także agentami, jest umiejscowiona w jednym lub wielu wierzchołkach grafu nazywanych bazami. Stamtąd poruszając się po z góry znanym (offline) lub nieznanym (online) grafie muszą wykonać powierzone im zadanie, takie jak przeszukanie grafu, spotkanie, dekontaminacja grafu czy wybór lidera. Celem jest znalezienie optymalnej, rozproszonej, deterministycznej strategii (sekwencji ruchów jednostek),...
    • Pełny tekst w portalu
  23. Równowaga strategiczna dla zbiorów defensywnych w drzewach

    W pracy rozważany jest problem defensywnej równowagi strategicznej dla zbiorów defensywnych w drzewach (spójnych grafach acyklicznych), który polega na znalezieniu dwóch rozłącznych globalnych zbiorów defensywnych. Zagadnienie to znajduje zastosowanie w modelo- waniu problemów komunikacyjnych w sieciach. Dla danego grafu G podzbiór jego wierzchołków S jest zbiorem defensywnym, jeśli dla każdego wierzchołka v należącego do S spełniony...
    • Pełny tekst w serwisie zewnętrznym
  24. Scheduling of identical jobs with bipartite incompatibility graphs on uniform machines. Computational experiments

    We consider the problem of scheduling unit-length jobs on three or four uniform parallel machines to minimize the schedule length or total completion time. We assume that the jobs are subject to some types of mutual exclusion constraints, modeled by a bipartite graph of a bounded degree. The edges of the graph correspond to the pairs of jobs that cannot be processed on the same machine. Although the problem is generally NP-hard,...
    • Pełny tekst w serwisie zewnętrznym
  25. Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines

    The problem of scheduling n identical jobs on 4 uniform machines with speeds s1>=s2>=s3>=s4 is considered.The aim is to find a schedule with minimum possible length. We assume that jobs are subject to mutual exclusion constraints modeled by a bipartite incompatibility graph of degree delta. We show that the general problem is NP-hard even if s1=s2=s3. If, however, delta<5 and s1>12s2 s2=s3=s4, then the problem can be solved to...
    • Pełny tekst w serwisie zewnętrznym
  26. Shared multi-processor scheduling

    We study shared multi-processor scheduling problem where each job can be executed on its private processor and simultaneously on one of many processors shared by all jobs in order to reduce the job’s completion time due to processing time overlap. The total weighted overlap of all jobs is to be maximized. The problem models subcontracting scheduling in supply chains and divisible load scheduling in computing. We show that synchronized...
    • Pełny tekst w serwisie zewnętrznym
  27. Szeregowanie zadań dwuprocesorowych w systemach otwartych

    W pracy rozważany jest problem szeregowania zadań dwuoperacyjnych w systemie otwartym (open-shop), z kryterium minimalizacji długości harmonogramu oraz sumy czasów zakończenia wszystkich zadań. Zakładając jednostkowe czasy wykonywania operacji można stosować efektywne metody chromatyczne rozwiązywania problemu, poprzez sprowadzenie go do modelu grafowego oraz zastosowanie w nim wybranego modelu kolorowania, które pozwala uzyskać...
    • Pełny tekst w serwisie zewnętrznym
  28. Szybkość przeszukiwania grafu

    R. Ostrowski - 2017
    Przeszukiwanie grafu pojawiło się jako problem matematyczny ponad 40 lat temu i w najogólniejszej wersji zajmuje się odszukiwaniem jednostki-uciekiniera niezależnie od jego poczynań. Od tamtej pory uzyskano wiele wyników odpowiadających na pytanie o minimalną ilość poszukujących jednostek w różnorodnych modelach, czyli odpowiednią liczbę przeszukiwawczą (ang. serach number) grafu. Popularne warianty problemów przeszukiwania obejmują...
  29. The Snow Team Problem

    D. Dereniowski , A. Lingas , M. Persson , D. Osula , P. Żyliński - 2017
    We study several problems of clearing subgraphs by mobile agents in digraphs. The agents can move only along directed walks of a digraph and, depending on the variant, their initial positions may be pre-specified. In general, for a given subset~$\cS$ of vertices of a digraph $D$ and a positive integer $k$, the objective is to determine whether there is a subgraph $H=(\cV_H,\cA_H)$ of $D$ such that (a) $\cS \subseteq \cV_H$, (b)...
    • Pełny tekst w serwisie zewnętrznym
  30. 2016

  31. An O ( n log n ) algorithm for finding edge span of cacti

    Let G=(V,E) be a nonempty graph and xi be a function. In the paper we study the computational complexity of the problem of finding vertex colorings c of G such that: (1) |c(u)-c(v)|>=xi(uv) for each edge uv of E; (2) the edge span of c, i.e. max{|c(u)-c(v)|: uv belongs to E}, is minimal. We show that the problem is NP-hard for subcubic outerplanar graphs of a very simple structure (similar to cycles) and polynomially solvable for...
    • Pełny tekst w serwisie zewnętrznym
  32. Applications of semi-definite optimization in quantum information protocols

    P. Mironowicz - 2016
    This work is concerned with the issue of applications of the semi-definite programming (SDP) in the field of quantum information sci- ence. Our results of the analysis of certain quantum information protocols using this optimization technique are presented, and an implementation of a relevant numerical tool is introduced. The key method used is NPA discovered by Navascues et al. [Phys. Rev. Lett. 98, 010401 (2007)]. In chapter...
  33. Bounds on the cover time of parallel rotor walks

    D. Dereniowski , A. Kosowski , D. Pająk , P. Uznański - JOURNAL OF COMPUTER AND SYSTEM SCIENCES - 2016
    The rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, a set of k identical walkers is deployed in parallel, starting from a chosen subset of nodes, and moving around the graph in synchronous steps. During the process, each node successively propagates walkers visiting it along its outgoing arcs in round-robin fashion, according to a fixed ordering. We consider...
    • Pełny tekst w serwisie zewnętrznym
  34. Distributed Evacuation in Graphs with Multiple Exits

    We consider the problem of efficient evacuation using multiple exits. We formulate this problem as a discrete problem on graphs where mobile agents located in distinct nodes of a given graph must quickly reach one of multiple possible exit nodes, while avoiding congestion and bottlenecks. Each node of the graph has the capacity of holding at most one agent at each time step. Thus, the agents must choose their movements strategy...
    • Pełny tekst w serwisie zewnętrznym
  35. Edge-coloring of 3-uniform hypergraphs

    We consider edge-colorings of 3-uniform hypergraphs which is a natural generalization of the problem of edge-colorings of graphs. Various classes of hypergraphs are discussed and we make some initial steps to establish the border between polynomial and NP-complete cases. Unfortunately, the problem appears to be computationally difficult even for relatively simple classes of hypergraphs.
    • Pełny tekst w serwisie zewnętrznym
  36. Eqiuitable coloring of corona products of cubic graphs is harder than ordinary coloring

    A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G. In this paper the problem of determinig the equitable coloring number for coronas of cubic graphs is studied. Although the problem of ordinary coloring of coronas...
    • Pełny tekst w serwisie zewnętrznym
  37. Equitable coloring of graphs. Recent theoretical results and new practical algorithms

    In this paper we survey recent theoretical results concerning conditions for equitable colorability of some graphs and recent theoretical results concerning the complexity of equitable coloring problem. Next, since the general coloring problem is strongly NP-hard, we report on practical experiments with some efficient polynomial-time algorithms for approximate equitable coloring of general graphs.
    • Pełny tekst w serwisie zewnętrznym
  38. Global defensive sets in graphs

    In the paper we study a new problem of finding a minimum global defensive set in a graph which is a generalization of the global alliance problem. For a given graph G and a subset S of a vertex set of G, we define for every subset X of S the predicate SEC ( X ) = true if and only if | N [ X ] ∩ S | ≥ | N [ X ] \ S | holds, where N [ X ] is a closed neighbourhood of X in graph G. A set S is a defensive alliance if and only if for...
    • Pełny tekst w serwisie zewnętrznym
  39. Increased Certification of Semi-device Independent Random Numbers using Many Inputs and More Postprocessing

    P. Mironowicz , A. Tavakoli , A. Hameedi , B. Marques , M. Pawłowski , M. Bourennane - NEW JOURNAL OF PHYSICS - 2016
    Quantum communication with systems of dimension larger than two provides advantages in information processing tasks. Examples include higher rates of key distribution and random number generation. The main disadvantage of using such multi-dimensional quantum systems is the increased complexity of the experimental setup. Here, we analyze a not-so-obvious problem: the relation between randomness certification and computational requirements...
    • Pełny tekst w serwisie zewnętrznym
  40. Independence in uniform linear triangle-free hypergraphs

    P. Borowiecki , M. Gentner , C. Löwenstein , D. Rautenbach - DISCRETE MATHEMATICS - 2016
    The independence number a(H) of a hypergraph H is the maximum cardinality of a set of vertices of H that does not contain an edge of H. Generalizing Shearer’s classical lower bound on the independence number of triangle-free graphs Shearer (1991), and considerably improving recent results of Li and Zang (2006) and Chishti et al. (2014), we show a new lower bound for a(H) for an r-uniform linear triangle-free hypergraph H with r>=2.
    • Pełny tekst w serwisie zewnętrznym
  41. Lossless Compression of Binary Trees with Correlated Vertex Names

    A. Manager , K. Turowski , W. Szpankowski - 2016
    Compression schemes for advanced data structures have become the challenge of today. Information theory has traditionally dealt with conventional data such as text, image, or video. In contrast, most data available today is multitype and context-dependent. To meet this challenge, we have recently initiated a systematic study of advanced data structures such as unlabeled graphs [1]. In this paper, we continue this program by considering...
    • Pełny tekst w serwisie zewnętrznym
  42. Network Graph Transformation Providing Fast Calculation of Paths for Resilient Routing

    Protection of transmission against failures can be appropriately dealt with by alternative paths. However, common schemes (e.g., Bhandaris scheme) are characterized by a remarkable delay while determining the transmission paths. This in turn may have a serious impact on serving dynamic demands (characterized by relatively short duration time). As a remedy to this problem, we introduce an approach to pre-compute the sets of disjoint...
  43. Non-isolating bondage in graphs

    A dominating set of a graph $G = (V,E)$ is a set $D$ of vertices of $G$ such that every vertex of $V(G) \setminus D$ has a neighbor in $D$. The domination number of a graph $G$, denoted by $\gamma(G)$, is the minimum cardinality of a dominating set of $G$. The non-isolating bondage number of $G$, denoted by $b'(G)$, is the minimum cardinality among all sets of edges $E' \subseteq E$ such that $\delta(G-E') \ge 1$ and $\gamma(G-E')...
    • Pełny tekst w serwisie zewnętrznym
  44. Normal-form preemption sequences for an open problem in scheduling theory

    B. Chen , E. Coffman , D. Dereniowski , W. Kubiak - JOURNAL OF SCHEDULING - 2016
    Structural properties of optimal preemptive schedules have been studied in a number of recent papers with a primary focus on two structural parameters: the minimum number of preemptions necessary, and a tight lower bound on shifts, i.e., the sizes of intervals bounded by the times created by preemptions, job starts, or completions. These two parameters have been investigated for a large class of preemptive scheduling problems,...
    • Pełny tekst w serwisie zewnętrznym
  45. On bipartization of cubic graphs by removal of an independent set

    H. Furmańczyk , M. Kubale , S. Radziszowski - DISCRETE APPLIED MATHEMATICS - 2016
    We study a new problem for cubic graphs: bipartization of a cubic graph Q by deleting sufficiently large independent set.
    • Pełny tekst w serwisie zewnętrznym
  46. On some open questions for Ramsey and Folkman numbers

    S. Radziszowski , X. Xiaodong - 2016
    We discuss some of our favorite open questions about Ramsey numbers and a related problem on edge Folkman numbers. For the classical two-color Ramsey numbers, we first focus on constructive bounds for the difference between consecutive Ramsey numbers. We present the history of progress on the Ramsey number R(5,5) and discuss the conjecture that it is equal to 43.
  47. On the hardness of computing span of subcubic graphs

    In the paper we study the problem of finding ξ-colorings with minimal span, i.e. the difference between the largest and the smallest color used.
    • Pełny tekst w serwisie zewnętrznym
  48. Sharp bounds for the complexity of semi-equitable coloring of cubic and subcubic graphs

    M. Kubale , H. Furmańczyk - 2016
    In this paper we consider the complexity of semi-equitable k-coloring of the vertices of a cubic or subcubic graph. We show that, given n-vertex subcubic graph G, a semi-equitable k-coloring of G is NP-hard if s >= 7n/20 and polynomially solvable if s <= 7n/21, where s is the size of maximum color class of the coloring.
    • Pełny tekst w serwisie zewnętrznym
  49. Strategic balance in graphs

    For a given graph G, a nonempty subset S contained in V ( G ) is an alliance iff for each vertex v ∈ S there are at least as many vertices from the closed neighbourhood of v in S as in V ( G ) − S. An alliance is global if it is also a dominating set of G. The alliance partition number of G was defined in Hedetniemi et al. (2004) to be the maximum number of sets in a partition of V ( G ) such that each set is an alliance. Similarly,...
    • Pełny tekst w serwisie zewnętrznym
  50. Szeregowanie identycznych zadań na czterech procesorach jednorodnych z dwudzielnymi grafami konfliktów

    M. Kubale - 2016
    Rozważono problem szeregowania n zadań jednostkowych na 4 procesorach jednorodnych o szybkościach s1>=s2>=s3>=s4. Celem szeregowania jest utworzenie najkrótszego możliwego harmonogramu. Zadania podlegają ograniczeniom zasobowym mówiącym, że niektóre pary zadań nie mogą być wykonane na tym samym procesorze. Podajemy algorytm dokładny, który rozwiązuje problem w czasie liniowym, o ile graf niezgodności jest kubiczny. Ponadto podajemy...
  51. Topology recognition and leader election in colored networks

    Topology recognition and leader election are fundamental tasks in distributed computing in networks. The first of them requires each node to find a labeled isomorphic copy of the network, while the result of the second one consists in a single node adopting the label 1 (leader), with all other nodes adopting the label 0 and learning a path to the leader. We consider both these problems in networks whose nodes are equipped with...
    • Pełny tekst w serwisie zewnętrznym
  52. 2015

  53. 2-outer-independent domination in graphs

    We initiate the study of 2-outer-independent domination in graphs. A 2-outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)\D has at least two neighbors in D, and the set V(G)\D is independent. The 2-outer-independent domination number of a graph G is the minimum cardinality of a 2-outer-independent dominating set of G. We show that if a graph has minimum degree at least two,...
    • Pełny tekst w serwisie zewnętrznym
  54. A bound on the number of middle-stage crossbars in f-cast rearrangeable Clos networks

    In 2006 Chen and Hwang gave a necessary and sufficient condition under which a three-stage Clos network is rearrangeable for broadcast connections. Assuming that only crossbars of the first stage have no fan-out property, we give similar conditions for f-cast Clos networks, where f is an arbitrary but fixed invariant of the network. Such assumptions are valid for some practical switching systems, e.g. high-speed crossconnects....