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			Home &gt; Graph Theory: Networking
					&gt; Shortest-Path Algorithms
					&gt; Bellman-Ford Algorithm</div>
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        <a class="toc_link" href="page21.xml">Designing A Routing Protocol</a>
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        <a class="toc_link" href="">Telephone Networks</a>
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        <a class="toc_link" href="page30.xml">Distance-Vector Routing</a>
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    <div id="content"><h1>Bellman-Ford Algorithm</h1>
				<p>
				The <i>Bellman-Ford algorithm</i> solves the
				single-source shortest-path problem.  It allows negative edge weights,
				but does not allow a directed cycle of negative weight.
				</p>
				<p>
				The Bellman-Ford algorithm returns <b>false</b>,
				indicating that no solution exists, if it encounters a cycle of
				negative weight that is reachable from the source. Otherwise, the
				algorithm returns <b>true</b>, indicating that it has
				found all shortest paths from the source.
				</p>
				
				

				<h2>The Algorithm</h2>
				
				<ol><li>
					<m:math>
						<m:mi>δ</m:mi>
						<m:mo>⁡</m:mo>
						<m:mfenced>
							<m:mi>s</m:mi>
							<m:mi>v</m:mi>
						</m:mfenced>
					</m:math>
					is the current estimate of the weight from vertex
					<m:math><m:mi>s</m:mi></m:math> to vertex <m:math><m:mi>v</m:mi></m:math>.
					</li><li>
					<m:math>
						<m:mo>π</m:mo>
						<m:mo>⁡</m:mo>
						<m:mfenced>
							<m:mi>v</m:mi>
						</m:mfenced>
					</m:math>
					is a predecessor of
					<m:math>
						<m:mi>v</m:mi>
						<m:mo>∈</m:mo>
						<m:mi>V</m:mi>
						<m:mo>⁡</m:mo>
						<m:mfenced>
							<m:mi>G</m:mi>
						</m:mfenced>
					</m:math>.
					In other words, the vertex through which the shortest distance from
					<m:math><m:mi>s</m:mi></m:math>
					to
					<m:math><m:mi>v</m:mi></m:math>
					was established.
					</li></ol>
				
				
				for each vertex
				<m:math>
					<m:mi>v</m:mi>
					<m:mo>∈</m:mo>
					<m:mi>V</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>G</m:mi>
					</m:mfenced>
					<m:mo>{</m:mo>
				</m:math>
				
				
				
				<m:math>
					<m:mi>δ</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>s</m:mi>
						<m:mi>v</m:mi>
					</m:mfenced>
					<m:mo>=</m:mo>
					<m:mn>∞</m:mn>
				</m:math>
				
				
				
				<m:math>
					<m:mi>π</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>v</m:mi>
					</m:mfenced>
					<m:mo>=</m:mo>
					<m:mi>nil</m:mi>
					<m:mo>;</m:mo>
					<m:mo>}</m:mo>
				</m:math>
				
				
				
				<m:math>
					<m:mi>δ</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>s</m:mi>
						<m:mi>s</m:mi>
					</m:mfenced>
					<m:mo>=</m:mo>
					<m:mn>0</m:mn>
					<m:mo>;</m:mo>
				</m:math>
				
				
				
				for
				<m:math>
					<m:mo>(</m:mo>
					<m:mi>k</m:mi>
					<m:mo>=</m:mo>
					<m:mn>1</m:mn>
					<m:mo>;</m:mo>
				
					<m:mi>k</m:mi>
					<m:mo>&lt;</m:mo>
					<m:mi>n</m:mi>
					<m:mo>;</m:mo>
				
					<m:mi>k</m:mi>
					<m:mo>+</m:mo>
					<m:mo>+</m:mo>
					<m:mo>)</m:mo>
				</m:math>
				
				
				
				for each edge
				<m:math>
					<m:mfenced>
						<m:mi>u</m:mi>
						<m:mi>v</m:mi>
					</m:mfenced>
					<m:mo>∈</m:mo>
					<m:mi>E</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>G</m:mi>
					</m:mfenced>
				</m:math>
				
				
				
				if
				<m:math>
					<m:mo>(</m:mo>
					<m:mi>δ</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>s</m:mi>
						<m:mi>v</m:mi>
					</m:mfenced>
				
					<m:mo>&gt;</m:mo>
				
					<m:mi>δ</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>s</m:mi>
						<m:mi>u</m:mi>
					</m:mfenced>
				
					<m:mo>+</m:mo>
				
					<m:mi>w</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>u</m:mi>
						<m:mi>v</m:mi>
					</m:mfenced>
					<m:mo>)</m:mo>
					<m:mo>{</m:mo>
				</m:math>
				
				
				
				<m:math>
					<m:mi>δ</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>s</m:mi>
						<m:mi>v</m:mi>
					</m:mfenced>
					<m:mo>=</m:mo>
					<m:mi>δ</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>s</m:mi>
						<m:mi>u</m:mi>
					</m:mfenced>
					<m:mo>+</m:mo>
					<m:mi>w</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>u</m:mi>
						<m:mi>v</m:mi>
					</m:mfenced>
					<m:mo>;</m:mo>
				</m:math>
				
				
				
				<m:math>
					<m:mi>π</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>v</m:mi>
					</m:mfenced>
					<m:mo>=</m:mo>
					<m:mi>u</m:mi>
					<m:mo>;</m:mo>
					<m:mo>}</m:mo>
				</m:math>
				
				
				
				for each edge
				<m:math>
					<m:mfenced>
						<m:mi>u</m:mi>
						<m:mi>v</m:mi>
					</m:mfenced>
					<m:mo>∈</m:mo>
					<m:mi>E</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>G</m:mi>
					</m:mfenced>
				</m:math>
				
				
				
				if
				<m:math>
					<m:mo>(</m:mo>
					<m:mi>δ</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>s</m:mi>
						<m:mi>v</m:mi>
					</m:mfenced>
					<m:mo>&gt;</m:mo>
					<m:mi>δ</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>s</m:mi>
						<m:mi>u</m:mi>
					</m:mfenced>
					<m:mo>+</m:mo>
					<m:mi>w</m:mi>
					<m:mo>⁡</m:mo>
					<m:mfenced>
						<m:mi>u</m:mi>
						<m:mi>v</m:mi>
					</m:mfenced>
					<m:mo>)</m:mo>
				</m:math>
				
				
				
				return <b>false</b>;
				
				
				
				return <b>true</b>;
				
				
				
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