COMP473: Advanced Algorithms

COMP473: Advanced Algorithms

Mark Dras

Department of ComputingMacquarie University

2007

Mark Dras (MacqU) COMP473: Advanced Algorithms 1 / 75

Insect-based Agents

Outline

1 Insect-based Agents

2 Ant Colony OptimisationTravelling SalesmanVertex CoverQuadratic Assignment ProblemTelecom NetworksGraph Partitioning


Insect-based Agents

Agents

different types of agents

we’re looking a low-level ones with simple behaviours

want the solution to emerge from interactions, rather than beingexplicitly programmed


Insect-based Agents

Game of Life

invented by John Conway, published in Martin Gardner’s“Mathematical Games” column in 1970

based on a 2D grid (usually a torus: wraparound edges) dividedinto cells

a cell has 8 neighbours, and can be alive or dead

Game of Life rules:

if a live cell has fewer than two live neighbours, it dies (of loneliness)if a live cell has more than three neighbours, it dies of overcrowdingif a dead cell has exactly three neighbours, it will come alive nextiteration


Insect-based Agents

Game of Life

look athttp://www.math.com/students/wonders/life/life.html

types of structures that appear:

‘still life’ (stable objects): block, beehive, boat, ship, loaf‘oscillators’ (periodic objects): blinker, toad

not all structures are easily predictable

r-pentamino: 5 adjacent squaresdoesn’t become stable until turn 1103


http://www.math.com/students/wonders/life/life.html

Insect-based Agents

Emergence

unexpected / unprogrammed behaviour occurs at the global level

you only program the low-level behaviour, but some interestinghigh-level or overall behaviour is producedin r-pentaminos there are complex, self-organising patterns thataren’t explicitly programmedthese patterns emerge from a lower-level specification


Insect-based Agents

Emergence

can be seen in economics

Adam Smith’s ‘invisible hand’economy just made up of individuals interested in their own income,consumption and savingbut at the macro level, economy moves toward equilibrium as ifguided by invisible hand

can be seen in ecology

sine behaviour of predator-prey is (maybe) emergent


Insect-based Agents

Emergence

two definitions of emergence

position 1: “emergence is just a word, like ‘random’, to cover upthe fact that we don’t know the underlying relationships”

position 2: “emergence is a kind of process in which the systemreturns robustly to an attractor that is an irresistible mathematicalnecessity of the dynamics that define it”


Insect-based Agents

Cellular Automata

a generalisation of the Game of Life

definition

a cell can have a value from a given set (say, 0 or 1)its state at time t + 1 is determined by the state of its neighboursand itself at time t

a 1D CA is just a string

rules define how substrings change

relationship to L-systems, dragon curves, etc


Insect-based Agents

Cellular Automata

rules for a 1D CA

1 1 1 → 01 1 0 → 11 0 1 → 01 0 0 → 10 1 1 → 10 1 0 → 00 0 1 → 10 0 0 → 0

consider string 01001100011011110111

can define different size neighbourhoods, different dimensions ofCA, different values for state


Insect-based Agents

Key Idea #1: Self-Organisation

structures appear at the global level of a system from interactionsamong its lower-level components

rules specifying interaction among lower-level components arepurely local

relies on four basic ingredients

positive feedbacknegative feedbackamplification of fluctuationsmultiple interactions


Insect-based Agents


1. positive feedback

bees can transmit information about a nectar source by dancingbee can choose to dance to indicate direction and distance tosource; or can continue foraging, or abandon food source andbecome an uncommitted followerhas been shown experimentally that a bee has a relatively highprobability of dancing for a good source and of abandoning a badonekey ideas are recruitment (getting other bees to come by dancing)and reinforcement (increasing the effect through multiple bees)similar story for ants laying pheromone trails


Insect-based Agents


2. negative feedback

necessary to avoid just fixating on first found solutionin general can stem from saturation, exhaustion, competitionin bee or ant foraging, can stem from limited number of availableforagers, satiation, food source exhaustion, crowding at the foodsource, competition between food sources


Insect-based Agents


3. amplification of fluctuations

random walks, errors, random task-switching, etchelps insects to avoid getting stuck in locally but not globally bestsolution


Insect-based Agents


4. multiple interactions

not just one single bee dancing one single time


Insect-based Agents

Key Idea #2: Stigmergy

individuals can interact directly

by antennation, trophallaxis (food or liquid exchange), mandibularcontact, visual contact

individuals can interact indirectly

one modifies the environment and the other responds to the newenvironmentthis is a type of stigmergy

e.g. termites

don’t have an overall plan for building nestsdon’t communicate directly even locally regarding shape of nesta stimulating configuration triggers the response of a termiteworker: cyclic


Insect-based Agents

Ant Systems

inspired by various ant behaviours

foragingdivision of labour and task allocationbrood sorting and cemetary organising

uses key idea of pheromone for stigmergy

this is a chemical that ants laycan accumulate and evaporate


Ant Colony Optimisation

Outline




Ant Colony Optimisation Travelling Salesman

Outline





TSP: Ant System

the earliest version (1992)

ants build a solution by moving on the problem graph from one cityto another until they complete a tour

during an iteration of the AS algorithm each ant k (k = 1 . . .m)builds a tour executing n = |V | steps in which a probabilistictransition rule is applied

iterations are indexed by t , 1 ≤ t ≤ tmax



TSP: Ant System

for each ant, the transition from city i to city j depends on:1 whether or not the city has already been visited: ants keep a record

(TABU LIST), which is emptied between tours2 VISIBILITY, defined as the inverse of the distance, ηij = 1/dij ;

represents the heuristic desirability of choosing city j when in city i ,and is static

3 amount of virtual PHEROMONE TRAIL τij(t); represents the learneddesirability of choosing city j when in city i



TSP: Ant System

random proportional TRANSITION RULE for ant k is given by

pkij (t) =

[τij(t)]α ·[ηij ]β

P

l∈Jki[τil (t)]α·[ηil ]β

if j ∈ Jki

0 otherwise

whereJk

i is the set of cities left to visit when at city iα, β are weighting parameters

if α = 0, the rule corresponds to the simple greedy algorithm; ifβ = 0, only pheromone amplification is at work, and initial(probably non-optimal) candidates will just be reinforced



TSP: Ant System

after completing a tour, each ant k lays a quantity of pheromone∆τ k

ij (t) on each edge (i , j) it has used, the amount depending onhow it has performed, giving the PHEROMONE LAYING RULE

∆τ kij (t) =

{

Q/Lk (t) if (i , j) ∈ T k(t)0 otherwise

whereT k(t) is the tour done by ant k at iteration tLk(t) is the length of T k(t)Q is a parameter (same order of magnitude as optimal tour length)



TSP: Ant System

pheromones also decay to avoid stagnation, giving the followingGLOBAL PHEROMONE UPDATE RULE

τij(t + 1)← (1− ρ) · τij(t) + ∆τij(t)

whereρ is the coefficient of decay, 0 ≤ ρ < 1∆τij(t) =

∑mk=1 ∆τ k

ij (t)m is the number of ants

number of ants m is also an important parameter: too many wouldreinforce suboptimal trails, too few would not lay enoughpheromone to find good paths



TSP: Ant System

original results weren’t so good

worked comparably with general heuristics (simulated annealing,tabu search), but outperformed by specialised heuristics

however, it worked differently from other heuristics

it keeps track of (with non-zero levels of pheromone) alternativegood solutionstypically, it doesn’t fix on one optimal solution: it will shuffle betweenseveralthis suggests it could be useful for dynamic problems



TSP: Ant Colony System (ACS)

based on four modifications of Ant System:

a different transition rulea different pheromone trail update rulethe use of local updates of pheromone trail to favour explorationthe use of a candidate list to restrict the choice of the next city tovisit



TSP: ACS

transition rule modified to allow explicitly for exploration

an ant k chooses a city j to move to following the rule

j =

{

arg maxu∈Jki

{

[τiu(t)] · [ηiu]β}

if q ≤ q0

J otherwise

whereq is uniform random over [0,1]q0 is a tunable parameter, 0 ≤ q0 ≤ 1J is a city randomly selected according to a given probability



TSP: ACS

probability for determining city J:

pkiJ(t) =

[τiJ(t)]·[ηiJ ]βP

l∈Jki[τil(t)]·[ηil ]β

if j ∈ Jki

0 otherwise

note that this is very similar to Ant System’s rule

q ≤ q0 corresponds to exploitation of available knowledgeq > q0 corresponds to more exploration

tuning of q0 corresponds to tuning of temperature in simulatedannealing



TSP: ACS

pheromone trail update modified by restricting global trail updateonly to best ant

in addition, whereas in Ant System pheromone was added to alledges, in ACS it is added only to edges in best tour since start oftrial

τij(t + 1)← (1− ρ) · τij(t) + ∆τij(t)

where(i , j)’s are those edges belonging to T +, the optimal tour∆τij(t) = 1/L+

L+ is the length of T +



TSP: ACS

there is also a local update of pheromone trail: when, whileperforming a tour, ant k is in city i and selects city j ∈ J k

i , thepheromone concentration of (i , j) is updated by the formula

τij(t)← (1− ρ) · τij(t) + ρ · τ0

whereτ0 is the initial value of pheromone trails (experimentally found tobe usefully set to τ0 = (n · Lnn)

1, where n is the number of citiesand Lnn is the optimal tour produced by the nearest neighbourheuristic)

this means that when an ant visits an edge, pheromonediminishes, encouraging exploration



TSP: ACS Results

tested on TSPLIB graphs

results by best tour length and number of iterations

ACS ACS GA GA EP EP SA SAbest # iter best # iter best # iter best # iter

Eil50 425 1830 428 25000 426 100000 443 68512Eil75 535 3480 545 80000 542 325000 580 173250KroA100 21282 4820 21761 103000 n/a n/a n/a n/a



TSP: ACS modifications

it’s been demonstrated that heuristic methods generally need toincorporate a local optimiser

this takes the solution at the end of a tour and drives it to its localoptimum

ACS combined with local search (3-opt)

results are compared with winner of First International Contest onEvolutionary Optimization, and are quite comparable



Local search: n-opt

in 2-opt, a 2-opt (or 2-exchange) neighbourhood of a candidatesolution s is defined as the set of all candidate solutions s ′ that canbe obtained by exchanging two pairs of edges in all possible ways

e.g. taking edges (x1, x2) and (x3, x4), s′ then consists of all of theedges in s but replacing the two selected by (x1, x3) and (x2, x4)

in 3-opt, at most three edges are replaced

in 2.5-opt, a 2-opt move is performed, and at the same time theinsertion of a vertex is evaluated


Ant Colony Optimisation Vertex Cover

Outline





VC: Ant Colony Optimization

from Shyu et al (2001)

question: how much tailoring is needed to apply the ant systemideas to another problem?

this is the reason ant systems are generally referred to as ameta-heuristic

the major difference with TSP (in terms of ant movement) is thatthe set of nodes constituting the VC isn’t guaranteed to form apath

therefore, from the graph G = 〈V ,E〉 we construct the completegraph Gc = 〈V ,Ec〉, where Ec = {(i , j) | i , j ∈ V}



VC: ACO

note that we still need to be able to reconstruct the original graphfrom Gc, so we define a function

ψk (r , j) =

{

1, if (r , j) ∈ E ;0, if (r , j) ∈ Ec − E

the function ψ is also used by the ants when deciding whether toput a vertex in the VC

the second major difference is that we want to select a subset ofvertices rather than of edges

hence, in pheromone laying, pheromone is placed on verticesrather than on edges

other differences are just in the specific definition of behaviourrules



VC: ACO

the transition rule is similar to TSP

pkj =

1, if q < q0 and j = arg maxr∈Ak{τrη

βrk};

0, if q < q0 and j 6= arg maxr∈Ak{τrη

βrk};

τjηβ

jkP

r∈Akτr η

β

rk

, if q ≥ q0,

whereAk denotes the set of accessible vertices for ant k ,ηjk is the heuristic goodness of node j ,τj is the amount of pheromone on node j ,β is a rational number greater than zero.



VC: ACO

as noted before, heuristic goodness will depend on whether theedges are ‘real’ or not

ηjk =

∑

(r ,j)∈Ecψk (r , j)

w(j).

note that this heuristic is dynamic

because each time we place a node j in the vertex cover we set forall r , ψk (r , j) = 0, the heuristic goodness for all adjacent nodes rchanges (cf. static heuristic goodness for TSP)



VC: ACO

the global update rule (again, for pheromone decay)

τi =

{

(1− ρ)τi + τ0ρ∆τi , if i ∈ V ′c

(1− ρ)τi , otherwise

where∆τi = 1

P

j∈V ′c

w(j) ,

V ′c are the nodes in the current best solution.



VC: ACO

the local update rule

τi = (1− ϕ)τi + ϕτ0

whereτ0 = |V |∗(|V |−|V ′

c |)P

j∈V ′c

w(j)

ϕ is an evaporation rate specific to the local update, 0 ≤ ϕ < 1



VC: ACO Results

the comparison is with a simple approximation algorithm, greedydegree; simulated annealing; and tabu search

note that Shyu et al had weighted nodes, so the problem is slightlymore general; numbers here represent total weight

n m GD SA TS ACO OPT10 30 449.0 451.8 451.8 431.3 431.315 60 818.5 763.4 763.4 729.0 726.220 80 987.7 927.9 927.9 905.4 898.025 100 1236.9 1148.0 1148.0 1107.4 1106.9



VC: ACO Results

in addition, times for ACO were significantly faster (e.g. for 25vertex instance, 0.02 CPU seconds vs 0.08 for SA and 0.078 forTS)

also, scales up to at least moderate-sized problems (300 vertices),in terms of both accuracy and speed


Ant Colony Optimisation Quadratic Assignment Problem

Outline





Quadratic Assignment Problem

what is it?

a set of n activities has to be assigned to n locationsthere is an n × n matrix D of distances dij between locationsthere is an n × n matrix of flows fhk between activity h and activity kan assignment is a permutation π of {1, . . . , n}, where π(i) is theactivity assigned to location iobjective function:

C(π) =

n∑

i,j=1

dij fπ(i)π(j)

the optimal permutation is given by:

πopt = arg minπ∈Π(n)

C(π)



QAP Characteristics

was originally formulated (1957) to describe assignment offacilities to locations, with the flows being things like transfers ofdata, material, people, . . .

also used for Steiner Wiring Problem (placing components on acomputer backboard so that total length of interconnected wiring isminimised)

also NP-hard

“harder” than e.g. vertex cover

any α-approximation solution is also NP-completeproblems where n > 15 are intractable for exact solutions, evenwhen using cutting planes or branch-and-bound



QAP Example

consider a randomly generated problem of order 4:

D =

0 10 4 210 0 6 4

4 6 0 12 4 1 0

,F =

0 3 8 33 0 2 48 2 0 53 4 5 0

consider also, of the 4! permutations, the following two:

π1 = (1)(2)(3)(4)π2 = (12)(34)



QAP Example

C(π1) = d1,1f1,1 + d1,2f1,2 + d1,3f1,3 + d1,4f1,4+d2,1f2,1 + d2,2f2,2 + d2,3f2,3 + d2,4f2,4+d3,1f3,1 + d3,2f3,2 + d3,3f3,3 + d3,4f3,4+d4,1f4,1 + d4,2f4,2 + d4,3f4,3 + d4,4f4,4+

= 202

C(π2) = d1,1f2,2 + d1,2f2,1 + d1,3f2,4 + d1,4f2,3+d2,1f1,2 + d2,2f1,1 + d2,3f1,4 + d2,4f1,3+d3,1f4,2 + d3,2f4,1 + d3,3f4,4 + d3,4f4,3+d4,1f3,2 + d4,2f3,1 + d4,3f3,4 + d4,4f3,3+

= 210

also have to check other 22 permutations to get lowest



QAP: Ant System

graph representation is similar to TSP

pheromone, representing the learned desirability of settingπ(i) = j , is also deposited on edges

need to derive a definition for heuristic goodness

consider vectors d̄ (‘distance potentials’) and f̄ (‘flow potentials’)obtained by

di =n

∑

j=1

dij {i = 1, . . . , n}

fh =n

∑

k=1

fhk {h = 1, . . . , n}

potentials represent total distance from a given node, and total flowfrom a given node; we’re interested in ‘barycentric nodes’ (smallesttotal distance: most central) and nodes with the greatest flow



QAP: Ant System Example

from previous example, vectors of potential are:

d =

1620117

,F =

149

1512

location 4 is central, 2 is peripheral; activity 3 is the most active, 2is least

basic heuristic, min-max coupling rule: attach high flow potentialsto low distance potentials



QAP: Ant System Example

construct coupling matrix B = d̄ · f̄ T to define heuristic goodness

B =

224 144 240 192280 180 300 240154 99 165 13298 63 105 84

apply min-max coupling to get bolded values

activity 3 w location 4; 1 w 3; 4 w 1; 2 w 2



QAP: Ant System

heuristic goodness is then ηij = bij = di · fjthe rest is the same as for TSP:

pkij (t) =

[τij(t)]α ·[ηij ]β

P

l∈Jki[τil (t)]α·[ηil ]β

if j ∈ Jki

0 otherwise

where Jki is the set of feasible activities (i.e. the set of activities

ant k has yet to assign)

difference with TSP is that ant starts at location with lowestdistance potential, and probabilistically chooses the best activity;then at the next step the second best location is chosen, etc


Ant Colony Optimisation Telecom Networks

Outline





Telecommunications Networks

basic problem: routing

a mechanism that allows information transmitted over a network tobe directed from a source to a destination through a sequence ofintermediate switching/buffering nodesnecessary because a fully connected network is too costlyaim is to direct traffic in a way that maximises network performancewhile minimising costs (e.g. rate of call rejection, throughput,packet delay, . . . )

previous problems have been static

network is both spatially distributed and dynamic: nodes or linkscan fail or become congested



Networks: Ant-Based Control

an adaptive routing algorithm based on ants that modify therouting policy at every node in a telephone network by depositingvirtual pheromone trail on routing table entries

ants’ goal is to build, and adapt to load changes at run time, routingtables so that network performance is maximisedperformance is measured by rate of incoming calls which areaccepted (calls can be rejected depending on availability ofresources at call time)



Networks: ABC

telephone network is represented by a graph with N nodes and Ebidirectional links

each node i has ki neighbours, and is characterised by capacity Ci ,spare capacity Si , and a routing table Ri = [r i

n,d (t)]ki ,N−1 (i.e. ki

rows, N − 1 cols)links have infinite capacityCi represents the maximum number of connections node i canestablish, and Si represents the percentage of capacity stillavailable for new connectionsrows in R are neighbour nodes and cols are destinationsthe value r i

n,d (t) is a probability that a given ant, whose destinationis node d , be routed from node i to neighbour node n



Networks: ABC

ants are launched, and move, probabilistically

the ant is deleted when it reaches its destinationa node’s routing table is updated after an ant visits it: pheromone isdeposited corresponding to the neighbour node the ant came from,and from its source node

calls (unlike ants) use routing tables deterministically: at set-uptime a route from node s to node d is built by choosingsequentially the neighbour node with the highest score(probability)

then decrease capacity of each node on route by fixed amount

calls are rejected if a node in the route under construction has noremaining capacity



Networks: ABC

an ant, whose destination is node 2 and whose source is node 5, arrives at node4 from node 5

col 2 gives ant probabilities of movement; col 5 is updated

/. -,() *+2

��

�??

???

/. -,() *+1

????

?

/. -,() *+3

/. -,() *+5

��

�

/. -,() *+4

1 2 3 51 0.8 0.3 0.1 0.13 0.1 0.4 0.8 0.15 0.1 0.3 0.1 0.8



Networks: ABC

math definitions: let s be the ant source node, i − 1 be the node it justcame from, i the current node at time t

entry r ii−1,s is reinforced while other entries r i

n,s(t), n 6= i − 1, in the samecol decay

r ii−1,s(t + 1) =

r ii−1,s(t) + δr

1 + δr

r in,s(t + 1) =

r in,s(t)

1 + δr, n 6= i − 1

where δr is a reinforcement parameter depending on ant characteristicssuch as age

in the previous slide’s example, the updated probabilities for node 5 aretherefore 0.1/(1 + δr), 0.1/(1 + δr), (0.8 + δr)/(1 + δr)



Networks: ABC

comments:

important assumption: any specific path connecting s and d mustexhibit approx the same level of congestion in both directions; OKfor telephone, questionable for packet-switchingamount of reinforcement δr depends on how well ant is performing;can use age T as a proxy (as delay ages ants)

δr =aT

+ b

age is manipulated by imposing a delay D on ants based on sparecapacity S

D = c · ed ·S

random noise is also added to avoid paths being ‘frozen’



Networks: ABC

five comparisons

fixed routing scheme based on shortest pathalgorithm based on mobile agents by Appleby and Stewardan improved version of mobile agentsABCABC with noise (works like the q0 parameter of ACS)



Networks: ABC

avg call fail std devshortest path 12.57 2.16mobile agents 9.19 0.78improved mobile agents 4.22 0.77ABC without noise 1.79 0.54ABC with noise 1.99 0.54



Networks: AntNet

differences from ABC:

the way ants collect and distribute information was designed so thatAntNet can be applied to both connection-oriented andconnectionless networks“ants collect information which is used to build local parametricmodels of the network status”, which are used to computereinforcements to change probabilistic routing tablesAntNet has been tested on a packet-switching network model, andits performance (throughput and delivery delays) has beencompared with well-known routing algorithmstest model based on continuous-time discrete event simulation,rather than sequential discrete iterations simulation



Networks: AntNet

uses same characteristics as ABC: s, d , Ri = [r in,d(t)]ki ,N−1 (the

latter thought of as desirability of a node)

in addition, define a set of estimates for trip times from currentnode i to all other nodes d (averages and variances)

Γi = {µi→d , σ2i→d}

two types of ants, FORWARD ANTS Fs→d and BACKWARD ANTS

Bd→s

forward ants move from source to dest at the same priority as datapackets (hence experience queues and traffic load)backward ants move in reverse, and backpropagate info alongcorresponding forward ant’s path, using high-priority queues



Networks: AntNet

forward ants Fs→d are launched at regular intervals from everynode s to random d

hops to neighbours are chosen randomly and proportionally todesirability, storing memory on a stackif a cycle is detected, cycle’s nodes are popped from ant’s stackand memory is destroyedwhen Fs→d reaches d , it generates a backward ant Bd→s and dies



Networks: AntNet

Bd→s follows path of Fs→d

updates Γi with (good) elapsed times stored in stack memoryupdates Ri by incrementing r i

i−1,d (t) and decreasing other r in,d (t)

increments r based on trip times Ti→d , best trip time Wi→d ,‘correcting term’

r ∝ (1−Wi→d/Ti→d) + ∆(σ,W )

other r in,d (t) decremented to normalise probabilities



Networks: AntNet

results compared with

open shortest path first (OPSF), current official Internet routingalgorithmsophisticated version of asynchronous distributed Bellman-Fordalgorithm (BF) with dynamic cost metricshortest path first (SPF) with dynamic cost metricQ-routing algorithm of Boyan and Littmanpredictive Q-routingdaemon, which knows instantaneously the states of all queues ofall nodes (as upper bound)

better than all, quite close to daemon


Ant Colony Optimisation Graph Partitioning

Outline





Graph Partitioning

previous problems used foraging metaphor from ants

this one uses ability of ants to sort

seen in brood sorting (sorting out different larvae into several piles)or cemetery organisationagain, building is by recognising local configurations



Graph Partitioning

problem is as presented in Bonabeau et al

start with an undirected unweighted graph G = 〈V ,E〉,V = {vi | i = 1, . . . ,N}, with adjacency matrix A = [aij ]

each vertex is initially assigned a set of random coordinates in Rn

the task is to change the vertices’ coordinates in Rn such that

1 clusters (cliques?) present in the graph are located in the sameportion of space

2 intercluster edges are minimised3 different clusters are clearly separated



Graph Partitioning

not sure what problem this is (not minimum k -clustering, notminimum clique partition, not . . . )

this partitioning is one way of laying out graphs



Graph Partitioning

definition of typical solution approach:

let (xim)m=1,...,n be the coordinate of vertex i in Rn, and

X = [xim](i=1,...,N)(M=1,...,n) the N × n matrix describing thecoordinates of all vertices

want to minimise quadratic cost function

C =

N∑

i=1

N∑

j=1

n∑

m=1

aij(xim − xjm)

under the constraint X T X = Inrelies heavily on matrix diagonalisation, which can be costlycomputationally



GP: Ant-based model

embedded in a plane

vertices move in the plane and are attracted to or rejected byclusters

distance metric is

d(vi , vj ) =|D(ρ(vi ), ρ(vj ))|

|ρ(vi )|+ |ρ(vj )|

whereρ(vi) = {vj ∈ V ; aij 6= 0} ∪ {vi}D(A,B) is the symmetric difference of two sets A,B

when two vertices vi , vj have many common adjacent nodes, D issmall, hence also d



GP: Ant-based model

also need to define probabilities of picking up and depositingvertices; assume agent is located at site r at time t and findsvertex vi

‘local density’ is

f (vi) =

(

1s2

P

vj∈Neigh(s×s)(r)

[1 − d(vi , vj)] if f > 0

0 otherwise

if all s2 sites around r are occupied, then f (vi) = 1 and the vertexshould not be picked up; if none are occupied, f (vi) = 0

picking up and dropping probabilities are

pp(vi ) =

„

k1

k1 + f (vi )

«2

, pd(vi ) =

„

f (vi )

k2 + f (vi )

«2



GP: Ant-based model

at time of book, no quantitive comparison available

for visual, check out Fig 4.10 in Bonabeau et al



Criticisms of ACO

from Pablo Moscato (of Memetic Algorithms)

need to solve two problems

original problemsetting of parameters

‘curse of dimensionality’

trying to guide search by sampling some kind of underlying optimaldistributionalso, near-optimal solutions may have very similar costs


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COMP473: Advanced Algorithms