figure 11.1 an 8-puzzle problem instance: (a) initial ...karypis/parbook/figures/chap11.pdf · 9 1...
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(a)
Blank tile
(b)
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(c)
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Figure 11.1 An 8-puzzle problem instance: (a) initial configuration; (b) final configuration; and (c)a sequence of moves leading from the initial to the final configuration.
Non-terminal node
Terminal node (goal)
Terminal node (non-goal)
x1 = 0 x1 = 1
x2 = 0 x2 = 1
x3 = 0 x3 = 0x3 = 1 x3 = 1
x4 = 0x4 = 0 x4 = 1x4 = 1
f (x) = 0 f (x) = 2
Figure 11.2 The graph corresponding to the 0/1 integer-linear-programming problem.
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Figure 11.3 Two examples of unfolding a graph into a tree.
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The last tile moved
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Step 3
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Figure 11.4 States resulting from the first three steps of depth-first search applied to an instanceof the 8-puzzle.
(c)(a) (b)
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Bottom of the stack
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Figure 11.5 Representing a DFS tree: (a) the DFS tree; successor nodes shown with dashed lineshave already been explored; (b) the stack storing untried alternatives only; and (c) the stack storinguntried alternatives along with their parent. The shaded blocks represent the parent state and theblock to the right represents successor states that have not been explored.
(a)
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(b)
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Figure 11.6 Applying best-first search to the 8-puzzle: (a) initial configuration; (b) final configura-tion; and (c) states resulting from the first four steps of best-first search. Each state is labeled withits h-value (that is, the Manhattan distance from the state to the final state).
C E F
BA
(a) (b)
D
Figure 11.7 The unstructured nature of tree search and the imbalance resulting from static parti-tioning.
Service any pending
messages
Do a fixed amount of work
Select a processor and
request work from it
Service any pending
messages
Finished
available
work
Got
work
Issued a request
Got a reject
Processor idle
Processor active
Figure 11.8 A generic scheme for dynamic load balancing.
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Current State
Figure 11.9 Splitting the DFS tree in Figure 11.5. The two subtrees along with their stack repre-sentations are shown in (a) and (b).
Step 1 Step 3
Step 5
Step 2
Step 6Step 4
w0 = 0.5
w0 = 0.5
w0 = 0.5
w0 = 0.25
w0 = 0.25
w0 = 1.0
w1 = 0.5 w1 = 0.5
w1 = 0.5 w1 = 0.25w1 = 0.25
w2 = 0.25 w2 = 0.25
w3 = 0.25
w3 = 0.25
Figure 11.10 Tree-based termination detection. Steps 1–6 illustrate the weights at various pro-cessors after each work transfer.
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ARRGRR
RP
p
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dup
Figure 11.11 Speedups of parallel DFS using ARR, GRR and RP load-balancing schemes.
GRRExpected (GRR)
RPExpected (RP)
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ber
of w
ork
requ
ests
Figure 11.12 Number of work requests generated for RP and GRR and their expected values(O(p log2 p) and O(p log p) respectively).
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E = 0.74E = 0.85E = 0.90
E = 0.64
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p log p2
Figure 11.13 Experimental isoefficiency curves for RP for different efficiencies.
Number of busyProcessors
Number of busyProcessors
Number of busyProcessors
Time
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(b)
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Time
Figure 11.14 Three different triggering mechanisms: (a) a high triggering frequency leads to highload-balancing cost, (b) the optimal frequency yields good performance, and (c) a low frequencyleads to high idle times.
Idle
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Global pointer
Global pointer
Figure 11.15 Mapping idle and busy processors with the use of a global pointer.
Expand the node to
generate successors
Expand the node to
generate successors
Expand the node to
generate successors
at designated processorGlobal list maintained
best node
nodes
Put expanded
Getcurrent
Pick the best node
from the list
Place generated
nodes in the list
Pick the best node
from the list
Place generated
nodes in the list
Unlock the list
Pick the best node
from the list
Place generated
nodes in the list
Unlock the listUnlock the list
Lock the list
Lock the list
Lock the list
P0
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Pp−1
Figure 11.16 A general schematic for parallel best-first search using a centralized strategy. Thelocking operation is used here to serialize queue access by various processors.
Exchangebest nodes
Exchangebest nodes
Exchangebest nodes
Local list
Local list
Local list
P0
P1
Pp−1
Figure 11.17 A message-passing implementation of parallel best-first search using the ring com-munication strategy.
Exchangebest nodes
Exchangebest nodesLocal list
Local list
Local list
blackboard
Exchangebest nodes
P0
P1
Pp−1
Figure 11.18 An implementation of parallel best-first search using the blackboard communicationstrategy.
Start node SStart node S
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Total number of nodes generated bysequential formulation = 13
Total number of nodes generated by
(b)
Goal node G Goal node G
two-processor formulation of DFS = 9
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R1
R3 L2
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L1R2
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Figure 11.19 The difference in number of nodes searched by sequential and parallel formulationsof DFS. For this example, parallel DFS reaches a goal node after searching fewer nodes than se-quential DFS.
L1
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Total number of nodes generated by
(a) (b)
Total number of nodes generated by
Start node S Start node S
Goal node GGoal node G
sequential DFS = 7 two-processor formulation of DFS = 12
Figure 11.20 A parallel DFS formulation that searches more nodes than its sequential counterpart.
x+2
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target = x+5
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Figure 11.21 Message combining and a sample implementation on an eight-processor hypercube.
(manager)
Processors
Search treeSubtasks
Subtask generator
Work request
Figure 11.22 The single-level work-distribution scheme for tree search.