[1][1][1][1] Lecture 4: Frequency reuse, channel assignment, and more June 15, Introduction to Algorithmic Wireless Communications David Amzallag

[2][2][2][2] A Reminder: Set Cover – Given a universe U of n elements, a collection of subsets of U, and a weight function. The problem is to find a minimum-weight subcollection of that covers all elements of U. NP – Is this problem NP-complete?

[3][3][3][3] How about the greedy algorithm? Greedy result: Optimal solution: Algorithm. While do effective Find the most effective set in the this iteration, S Pick S, and for each set.. Output the picked sets.

[4][4][4][4] Analysis – Number the elements of U in the order they were covered by the algorithm. Let be this numbering. – For each – The greedy algorithm is an -approximation algorithm for the minimum- weight set cover, where since

[5][5][5][5] Improvement of the performance guarantee tight example – Any tight example for this algorithm? NPP – Feige (1996) show that for any, if there is a factor approximation algorithm for the cardinality set cover problem, then NP=P.

[6][6][6][6] Frequency reuse revisited: cluster size and reuse distance D frequency group A frequency group B Cluster with cluster size N Co-channel cells

[7][7][7][7] The geometry of a hexagonal cell Unit scale is distance between neighboring cell centers. For cell radius convert To find the distance from the origin,, of point, convert axis:

[8][8][8][8] The geometry of a hexagonal cell [continue] – So, – Using this equation to locate co-channel cells, we start from a reference cell and move i hexagons along the u -axis and then j hexagons along the v -axis. – The distance between co-channel cells in adjacent clusters is – The number of cells in a cluster, N, is hence since i and j can only take integer values. frequency reuse factor – The frequency reuse factor, Q, is defined by

[9][9][9][9] Channel assignment strategies increasing capacityminimizing interference – For efficient utilization of the radio spectrum, a frequency reuse scheme is consistent with the objectives of increasing capacity and minimizing interference. fixed dynamic – A variety of channel assignment strategies has been developed: fixed or dynamic. – The choice impacts the performance of the system, particularly as to how calls are managed when a mobile is handed off from one cell to another.

[10] Optimal channel assignments is NP-hard, in general graph coloring – A special case is graph coloring. – For example, the graph for reuse 2. – Notice that coloring 3-colorable graph is NP-hard, in general. planar graphs – This is true also for planar graphs (Stockmeyer, 1973). – What is the graph for reuse 3?

[11] Fixed channel assignment – Each cell is allocated a predetermined set of channels. – Any call attempt within the cell can only be served by the unused channel. blocked – If all the channels in that cell are occupied, the call is blocked and the subscriber does not receive service. – Several variations of the fixed assignment strategy exists.

[12] Dynamic channel assignment – Channels are not allocated to different cells permanently. – Instead, each time a call request is made, the serving BS requests a channel from the RNC. – The RNC then allocates a channel to the requested cell following an algorithm that takes into account the likelihood of future blocking within the cell, the frequency of use of the candidate channel, the reuse distance of the channel, and other cost functions. – Accordingly, the RNC only allocates a given frequency if that frequency is not presently in use in the cell or any other cell which falls within the minimum restricted distance of frequency reuse to avoid co-channel interference.

[13] Dynamic channel assignment [continue] traffic distribution – Dynamic channel assignment strategies require the RNC to collect real-time data on channel occupancy, traffic distribution, and radio signal strength indications (RSSI) of all channels on a continuous basis. – This increases the storage and computational load on the system but provides the advantage on increased channel utilization and decreased probability of blocked call. Flexible channel assignment – There are also hybrid strategies (Flexible channel assignment).

[14] Rules for channel assignment – Do not assign co-channels or adjacent channels at the same cell site (not applicable for CDMA). – Do not assign co-channels in adjacent cell sites (not applicable for CDMA). – Do not mix and match channel groups in a cell or sector. – Avoid adjacent channel assignment in adjacent cell sites (not applicable for CDMA). – Maintain proper channel separation for any channel assignment for a sector or site. – Maximize the distance between reusing cell sites.

[15] Coloring the vertices of a graph coloring – A (proper) coloring of a graph G is an assignment of a color to each vertex so that adjacent vertices receive distinct colors. – Equivalently, it is partition of the vertices into independent sets. – The graph coloring problem is to color a given graph G with minimum number of colors,. k -colorable – A graph G with is called k -colorable. bipartite graphsplanar graphs cliques cycles – Examples: bipartite graphs are 2-colorable, planar graphs are 4-colorable, cliques are n-colorable, and even (odd) cycles are 2-colorable (3-colorable).

[16] Solutions, approaches, and results NP – The graph coloring problem ( MINIMUM CHROMATIC NUMBER ) is NP-hard. – The problem is approximable within (Halldórsson, 1993), and not approximable within for any (Bellare, Goldreich, Sudan, 1998). – Greedy heuristics: – “First fit” ( ) to DSATUR (Brelaz, 1979). –k -colorable graphs can be optimally solved using time (Eppstein, 2001). – Integer programming approaches.

[17] Coloring perfect graphs perfect – A graph G is perfect if for each induced subgraph H of G, – Another characterization – Another characterization. A graph is perfect iff neither nor contains an odd cycle of length at least 5 as an induced subgraph (Berge, 1966 and Chudnovski, Robertson, Seymour, Tomas, 2006). – Bipartite graphs are perfects. – Optimal coloring of a perfect graphs can be done in polynomial-time by using the Ellipsoid method (Grotschel, Lovasz, Schrijver, 1988).

[18] List coloring problem – graph coloring with constraints on the vertices – Given a graph with lists of permitted colors for each vertex – When can we color G (in the usual sense) so that each vertex receive color from its list? k -choosable – A graph G is called k -choosable, if, for every family with for all v, there is a vertex coloring of G from the lists choice number – The least integer k for which G is k -choosable is the choice number of G. – In general, for every G,

[19] Introducing the choice number of a graph – Even-length cycles are trivial example for graphs G with – How about graphs G with

[20] Every planar graph is 5-choosable (Thomassen, 1994) A stronger claim for planar graphs with at least 3 vertices: Suppose that – Every inner face is bounded by a triangle and its outer face by a cycle – has already been colored with color 1, and by color 2. – All other vertices of have lists of at least 3 colors. – All other vertices of have lists of at least 5 colors. Then the coloring of and can be properly extended. Proof is by induction on

[21] T-coloring problem – graph coloring with constraints on the edges – Given a graph with set T of nonnegative integers which represents disallowed separations between channels assigned to BS’s vertices. – T -coloring – A T -coloring of G is a coloring function such that – Special cases:

[22] What should be the objective function? – Minimizing the number of different colors – also known as T -chromatic number,, of G. span, – Minimizing the span,, i.e. the maximum of – Examples: 1. What are 2. What are NP – Computing and is NP-hard.

[23] Set coloring of reuse-2-hexagonal graphs

[24] Background – Given a graph with nonnegative demand on its vertices set coloring – A set coloring is an assignment to each vertex of distinct colors such that adjacent vertices receive disjoint set of colors. set chromatic number – The least value of the total number of color classes for which there is such coloring is the set chromatic number of set coloring problem – The set coloring problem is to find a set coloring using few colors as possible. – In our case, G is the reuse-2-hexagonal graph (triangular lattice). NP – Deciding whether is 3-colorable is NP-hard (McDiarmid, Reed, 2000).

[25] The concept of replicated graph – There is a natural graph associated with a pair, obtained by replacing each vertex v by a clique on vertices. replicated graph – We call, the replicated graph. – Set coloring of correspond to usual vertex coloring of, and the set chromatic number of is of – If a graph is perfect, then its replicated graph is also perfect, for any vector of demands. – What is the maximum size of a clique in – Any lower (upper) bound for the problem?

[26] Set coloring of bipartite graphs – First, find a proper 2-coloring of, partitioning the vertices into two sets A and B, and determine – To specify an optimal set coloring, use the colors – Color every with colors and every with colors