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    <p>The basic idea of B &amp; B is:</p> <ol> <li>When solving an optimisation problem ("Find an X satisfying criteria Y so as to minimise the cost f(X)"), you build a solution piece by piece -- at any point in time, you have a <em>partial solution</em>, which has a <em>cost</em>.</li> <li>If the nature of the problem is such that the cost of a partial solution can only stay the same or go up as you continue adding pieces to it, then you know that <strong>there's no point continuing to add pieces to a partial solution if there's already a full solution with lower cost</strong>. In this case, you can abandon (or "prune", or "fathom") further processing of this partial solution.</li> </ol> <p>Many problems have the latter property, making B &amp; B a widely applicable algorithm technique.</p> <p>The process of searching for solutions can be represented by a <em>search tree</em>, where the root node represents the starting point where no decisions have been made, and each edge leading from a node represents a decision about something to be included in a partial solution. Each node is a partial solution comprising the decisions made (edges) from the root to that node.</p> <p><strong>Example:</strong> if we want to solve a Sudoku puzzle, the root node would represent the board with just the originally supplied numbers filled in; there might be 9 edges from this root, each representing the decision to assign a number 1-9 to the top-left cell. Each of those 9 partial solution nodes could have 8 branches, representing the valid assignments to the cell at position (1, 2), and so on. Usually, each edge represents a recursion step in a program.</p> <p>With B &amp; B, in the best case a good solution is found early, meaning that unpromising areas of the search tree can be pruned near the root; but in the worst case, the entire tree of valid solutions will be generated. For this reason B &amp; B is usually only used to solve problems for which no faster algorithm is known (such as NP-hard problems).</p>
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