[[11]]

Lecture 4:Lecture 4: Frequency reuse, channel assignment, and Frequency reuse, channel assignment, and moremore

June 15, 2008

896960896960

Introduction to Algorithmic Introduction to Algorithmic Wireless CommunicationsWireless Communications

David AmzallagDavid Amzallagdavid.amzallag@bt.com david.amzallag@bt.com

www.cs.technion.ac.il/~amzallag/awcwww.cs.technion.ac.il/~amzallag/awc

[[22]]

A Reminder: Set CoverA 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.

– Is this problem NPNP-complete?

[[33]]

How about the greedy algorithm? How about the greedy algorithm?

Greedy result: Greedy result:

Optimal solution: Optimal solution:

Algorithm.Algorithm.

While do

Find the most effectiveeffective set in the this iteration, S

Pick S, and for each set .

.

Output the picked sets.

[[44]]

AnalysisAnalysis

– 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

[[55]]

Improvement of the performance guaranteeImprovement of the performance guarantee

– Any tight exampletight example for this algorithm?

– Feige (1996) show that for any , if there is a factor approximation algorithm for the cardinality set cover problem, then NPNP=PP.

[[66]]

Frequency reuse revisited: cluster size and reuse distanceFrequency reuse revisited: cluster size and reuse distance

DD

frequency group A

frequency group B

Cluster with cluster size N

Co-channel cells

[[77]]

The geometry of a hexagonal cellThe geometry of a hexagonal cell

Unit scale is distance between neighboring cell centers. For cell radius

To find the distance from the origin, , of point , convertconvert axis:

[[88]]

The geometry of a hexagonal cell The geometry of a hexagonal cell [continue][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.

– The frequency reuse factorfrequency reuse factor, Q, is defined by

[[99]]

Channel assignment strategiesChannel assignment strategies

– For efficient utilization of the radio spectrum, a frequency reuse scheme is consistent with the objectives of increasing capacityincreasing capacity and minimizing interferenceminimizing interference.

– A variety of channel assignment strategies has been developed: fixedfixed or dynamicdynamic.

– 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.

[[1111]]

Fixed channel assignmentFixed 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.

– If all the channels in that cell are occupied, the call is blockedblocked and the subscriber does not receive service.

– Several variations of the fixed assignment strategy exists.

[[1212]]

Dynamic channel assignmentDynamic 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.

[[1313]]

Dynamic channel assignment Dynamic channel assignment [continue][continue]

– Dynamic channel assignment strategies require the RNC to collect real-time data on channel occupancy, traffic distributiontraffic 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.

– There are also hybrid strategies (Flexible channel assignmentFlexible channel assignment).

[[1414]]

Rules for channel assignmentRules 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.

[[1515]]

Coloring the vertices of a graphColoring the vertices of a graph

– A (proper) coloringcoloring 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, .

– A graph G with is called kk-colorable-colorable.

– Examples: bipartite graphsbipartite graphs are 2-colorable, planar graphsplanar graphs are 4-colorable, cliquescliques are n-colorable, and even (odd) cyclescycles are 2-colorable (3-colorable).

[[1616]]

Solutions, approaches, and resultsSolutions, approaches, and results

– The graph coloring problem (MINIMUM CHROMATIC NUMBER) is NPNP-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.

[[1717]]

Coloring perfect graphsColoring perfect graphs

– A graph G is perfectperfect if for each induced subgraph H of G,

– Another characterizationAnother 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).

[[1818]]

List coloring problem – graph coloring with constraints on List coloring problem – graph coloring with constraints on the verticesthe 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?

– A graph G is called kk-choosable-choosable, if, for every family with for all v, there is a vertex coloring of G from the lists

– The least integer k for which G is k-choosable is the choice numberchoice number of G.

– In general, for every G,

[[1919]]

Introducing the choice number of a graphIntroducing the choice number of a graph

– Even-length cycles are trivial example for graphs G with

– How about graphs G with

{2,3}

{2,3}

{1,3}

{1,3}

{1,2} {1,2}

{1,2}{1,2}{1,2}

{1,2} {1,2}

{1,3}{1,3}

{2,3}{2,3}

{1,2} {3,4}

{2,4} {2,3} {1,4} {1,3}

{1,2}

{1,3}

{2,3}

{1,2}

{1,3}

{2,3}

[[2020]]

Every planar graph is 5-choosable (Thomassen, 1994)Every planar graph is 5-choosable (Thomassen, 1994)

1v

2v w

v

1

2

1G

2G

kv

1vkv

1kv

2u3u

'C

P

2v1u

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

[[2121]]

TT-coloring problem – graph coloring with constraints on the -coloring problem – graph coloring with constraints on the edgesedges

– Given a graph with set T of nonnegative integers which represents disallowed separations between channels assigned to BS’s vertices.

–

– A TT-coloring-coloring of G is a coloring function such that

– Special cases:

[[2222]]

What should be the objective function?What should be the objective function?

– Minimizing the number of different colors

– also known as T-chromatic number, , of G.

– Minimizing the spanspan, , , i.e. the maximum of

– Examples:

1.

What are

2.

What are

– Computing and is NPNP-hard.

5

41

23

1 4

4

17

[[2323]]

Set coloring of reuse-2-hexagonal graphsSet coloring of reuse-2-hexagonal graphs

[[2424]]

BackgroundBackground

– Given a graph with nonnegative demand on its vertices

– A set coloringset coloring is an assignment to each vertex of distinct colors such that adjacent vertices receive disjoint set of colors.

– The least value of the total number of color classes for which there is such coloring is the set chromatic numberset chromatic number of

– The set coloring problemset 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).

– Deciding whether is 3-colorable is NPNP-hard (McDiarmid, Reed, 2000).

[[2525]]

The concept of replicated graphThe concept of replicated graph

– There is a natural graph associated with a pair , obtained by replacing each vertex v by a clique on vertices.

– We call , the replicated graphreplicated 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?

[[2626]]

Set coloring of bipartite graphsSet 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