Heuristic Search
Blai Bonet
Breadth-first search and uniform-cost search
Universidad Sim´
on Bol´ıvar, Caracas, Venezuela
c 2015 Blai Bonet
Goals for the lecture
Lecture 5
Breadth-first search
Breadth-first search explores the search tree layer by layer expanding
all nodes at depth d before expanding any node at depth d + 1
• Learn breadth-first search and uniform-cost search
• Tree-search and graph-search variants of the algorithms
Nodes get ordered for expansion using a FIFO queue
• Fundamental characteristics
Algorithm can be implemented as a tree- or graph-search algorithm
depending on whether duplicates are pruned or not
• Perform analysis on canonical search trees
Breadth-first search is a fundamental algorithms in CS
c 2015 Blai Bonet
Lecture 5
c 2015 Blai Bonet
Lecture 5
Breadth-first search for explicit graphs [CLRS]
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Breadth-first search: example
breadth-first-search(Vertex root):
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% initialization
foreach Vertex u
color[u] = White
distance[u] = ∞
parent[u] = null
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Queue q
color[root] = Gray
distance[root] = 0
q.insert(root)
% FIFO queue
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% search
while !q.empty()
Vertex u = q.pop()
foreach Vertex v in adj[u]
if color[v] == White
color[v] = Gray
distance[v] = distance[u] + 1
parent[v] = u
q.insert(v)
color[u] = Black
c 2015 Blai Bonet
search tree
Lecture 5
c 2015 Blai Bonet
Breadth-first search: example
Lecture 5
Breadth-first search for implicit graphs
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% breadth-first search without duplicate elimination
breadth-first-search():
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% initialization
Queue q
Node root = make-root-node(init())
q.insert(root)
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% search
while !q.empty()
Node n = q.pop()
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% check for goal
if n.state.is-goal() return n
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% expand node
foreach <s,a> in n.state.successors()
q.insert(n.make-node(s,a))
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c 2015 Blai Bonet
Lecture 5
return null
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% failure: there is no path from root to goal
Lecture 5
Recall API and basic primitives
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Differences between breadth-first searches
struct AbstractState {
bool is-goal()
list<pair<State, Action> > successors()
}
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DerivedState init()
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CLRS’s breadth-first search is for explicit graphs, ours is for implicit
graphs and thus explores the search tree associated to a graph
struct Node {
Node(AbstractState s, Node p, Action a, unsigned gcost)
: state(s), parent(p), action(a), g(gcost)
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Our breadth-first search returns as soon as a node associated to a
goal state is selected for expansion (why not generated?)
Node make-node(AbstractState state, Action action)
return new Node(state, this, action, g + c(state, action))
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extract-path(vector<Action> &reversed-path)
Node node = this
while node != 0 && node.parent != 0
reversed-path.push-back(node.action)
node = node.parent
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}
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Node make-root-node(AbstractState state)
return new Node(state, null, -1, 0)
c 2015 Blai Bonet
Lecture 5
c 2015 Blai Bonet
Breadth-first search for implicit graphs
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Extended API for duplicate elimination I
% breadth-first search with duplicate elimination
breadth-first-search():
% initialization
Queue q
set-color(init(),Gray)
q.insert(make-root-node(init()))
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Lecture 5
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% search
while !q.empty()
Node n = q.pop()
void set-color(AbstractState state, Color color)
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unsigned get-color(AbstractState state)
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% check for goal
if n.state.is-goal() return n
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Later on we’ll see how to implement these functions efficiently
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% expand node
foreach <s,a> in n.state.successors()
if get-color(s) == White
set-color(s,Gray)
q.insert(n.make-node(s,a))
set-color(n.state,Black)
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return null
c 2015 Blai Bonet
% failure: there is no path from root to goal
Lecture 5
c 2015 Blai Bonet
Lecture 5
Properties of breadth-first search
Uniform-cost search
Expands all nodes at depth d before expanding any at depth d + 1
Breadth-first search finds shortest path rather than minimum-cost
path from initial state to goal state
– Complete: if there is a path, outputs a path, else outputs null
Uniform-cost search explores search tree cost-layer by cost-layer
expanding all nodes at cost c before expanding any node at cost > c
– Optimality: it returns a shortest path (not min cost!)
– Time complexity: O(bd ) (i.e. exponential in goal depth d)
Nodes get ordered for expansion using a priority queue
– Space complexity: O(bd ) (i.e. exponential in goal depth d)
Like breadth-first search, algorithm can be implemented as tree- or
graph-search depending on whether duplicates are pruned or not
Time and space complexities calculated in canonical search tree
with branching factor b where shallowest goal appears at depth d
c 2015 Blai Bonet
Lecture 5
c 2015 Blai Bonet
Uniform-cost search for implicit graphs
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Uniform-cost search: expansion
% uniform-cost search with duplicate elimination
uniform-cost-search():
% initialization
PriorityQueue q
% ordered by node’s g
set-color(init(),Gray)
set-distance(init(),0)
q.insert(make-root-node(init()))
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% search
while !q.empty()
Node n = q.pop()
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% check for goal
if n.state.is-goal() return n
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% expand node
expansion-for-uniform-cost-search(n,q)
set-color(n.state,Black)
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return null
c 2015 Blai Bonet
% is it the first time we see the state?
if get-color(s) == White
set-color(s,Gray)
set-distance(s,g)
q.insert(n.make-node(s,a))
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expansion-for-uniform-cost-search(Node n, PriorityQueue q):
foreach <s,a> in n.state.successors()
g = n.g + c(n.state,a)
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Lecture 5
% a shorter path was found, update open list
else if g < get-distance(s)
assert(get-color(s) == Gray)
% why?
set-distance(s,g)
s.node.parent = n
s.node.action = a
s.node.g = g
q.decrease-priority(s.node,g)
% failure: there is no path from root to goal
Lecture 5
c 2015 Blai Bonet
Lecture 5
Extended API for duplicate elimination II
Traveling in Romania (AIMA)
Oradea
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Neamt
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Zerind
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Iasi
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Arad
void set-color(AbstractState state, Color color)
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Sibiu
unsigned get-color(AbstractState state)
Vaslui
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void set-distance(AbstractState state, unsigned dist)
Rimnicu Vilcea
Timisoara
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Fagaras
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unsigned get-distance(AbstractState state)
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Lugoj
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Pitesti
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Mehadia
Later on we’ll see how to implement these functions efficiently
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Drobeta
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Hirsova
Urziceni
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Bucharest
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Craiova
Giurgiu
Eforie
Task: find optimal path from Rimnicu Vilcea to Vaslui
c 2015 Blai Bonet
Lecture 5
c 2015 Blai Bonet
Uniform-cost search: example
Lecture 5
Delayed duplicate elimination
Rimnicu Vilcea
Oradea
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Sibiu
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Arad
220 Fagaras
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Craiova
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Pitesti
Drobeta
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Bucharest
Often one implements a weaker form of duplicate elimination in
which duplicates are allowed to enter the open list (nodes generated
that are waiting for expansion)
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We call this form delayed duplicate elimination
Zerind
c 2015 Blai Bonet
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Timisoara
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Mehadia
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Lugoj
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Giurgiu
475 Urziceni
Vaslui
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It can be thought as that the search explicates graph edges rather
than graph vertices
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Eforie
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Lecture 5
c 2015 Blai Bonet
Lecture 5
UCS with delayed duplicate elimination
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Properties of uniform-cost search
% uniform-cost search with delayed duplicate elimination
uniform-cost-search():
PriorityQueue q
% ordered by node’s g
set-color(init(),Gray)
set-distance(init(),0)
q.insert(make-root-node(init()))
– Complete: if there is a path, outputs a path else outputs null (if
cmin > 0 or duplicates pruned)
– Optimality: it returns a minimum-cost path
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% search
while !q.empty()
Node n = q.pop()
State ns = n.state
– Time complexity: O(bc
– Space complexity: O(bc
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% delayed duplicate elimination
if get-color(ns) == White || n.g < get-distance(ns)
set-distance(ns,n.g)
if ns.is-goal() return n
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% expand node
foreach <s,a> in ns.successors()
set-color(s,Gray)
set-distance(s,n.g+c(ns,a))
q.insert(n.make-node(s,a))
set-color(ns,Black)
return null
% failure: there is no path from root to goal
c 2015 Blai Bonet
Lecture 5
Summary
• Breadth-first search
• Properties of breadth-first search
• Uniform-cost search
• Properties of uniform-cost search
• Delayed duplicate detection (search explicates edges rather than
nodes)
c 2015 Blai Bonet
) (= O(bd ) if all costs are 1)
∗ /c
min
) (= O(bd ) if all costs are 1)
If duplicates aren’t pruned and minimum edge cost cmin = 0,
uniform-cost search may not terminate (i.e. it’s not complete)
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∗ /c
min
Lecture 5
Time and space complexities calculated in canonical search tree
with branching factor b and cmin > 0 where minimum-cost goal
appears at depth d with cost c∗
c 2015 Blai Bonet
Lecture 5