As a rule, greedy algorithms are both intuitively appealing and simple. Given an optimization problem, it is usually easy to figure out how to proceed in a greedy manner, possibly after considering a few small instances of the problem.

Chapter **9**

**Greedy
Technique**

*Greed, for lack of a better word, is good! Greed is right! Greed
works!*

—Michael
Douglas, US actor in the role of Gordon Gecko, in the film *Wall Street*, 1987

Let us revisit the ** change-making
problem** faced, at least subconsciously, by millions of cashiers all
over the world: give change for a specific amount

Giving one more dime left you
with 3 cents to be given with three pennies.

Is this solution to the
instance of the change-making problem optimal? Yes, it is. In fact, one can
prove that the greedy algorithm yields an optimal solution for every positive
integer amount with these coin denominations. At the same time, it is easy to give
an example of coin denominations that do not yield an optimal solution for some
amounts—e.g., *d*_{1} = 25*,
d*_{2} = 10*, d*_{3} = 1 and ** n** = 30.

The approach applied in the
opening paragraph to the change-making prob-lem is called ** greedy**. Computer
scientists consider it a general design technique despite the fact that it is
applicable to optimization problems only. The greedy approach suggests
constructing a solution through a sequence of steps, each ex-panding a
partially constructed solution obtained so far, until a complete solution to
the problem is reached. On each step—and this is the central point of this
technique—the choice made must be:

*feasible*, i.e., it has to satisfy the problem’s constraints

*locally optimal*, i.e., it has to be the best local choice among all feasible
choices* *available on that step

*irrevocable*, i.e., once made, it cannot be changed on subsequent steps of the* *algorithm

These requirements explain
the technique’s name: on each step, it suggests a “greedy” grab of the best
alternative available in the hope that a sequence of locally optimal choices
will yield a (globally) optimal solution to the entire problem. We refrain from
a philosophical discussion of whether greed is good or bad. (If you have not
seen the movie from which the chapter’s epigraph is taken, its hero did not end
up well.) From our algorithmic perspective, the question is whether such a
greedy strategy works or not. As we shall see, there are problems for which a
sequence of locally optimal choices does yield an optimal solution for every
instance of the problem in question. However, there are others for which this
is not the case; for such problems, a greedy algorithm can still be of value if
we are interested in or have to be satisfied with an approximate solution.

In the first two sections of
the chapter, we discuss two classic algorithms for the minimum spanning tree
problem: Prim’s algorithm and Kruskal’s algorithm. What is remarkable about
these algorithms is the fact that they solve the same problem by applying the
greedy approach in two different ways, and both of them always yield an optimal
solution. In Section 9.3, we introduce another classic algorithm— Dijkstra’s
algorithm for the shortest-path problem in a weighted graph. Section 9.4 is
devoted to Huffman trees and their principal application, Huffman codes—an
important data compression method that can be interpreted as an application of
the greedy technique. Finally, a few examples of approximation algorithms based
on the greedy approach are discussed in Section 12.3.

As a rule, greedy algorithms
are both intuitively appealing and simple. Given an optimization problem, it is
usually easy to figure out how to proceed in a greedy manner, possibly after
considering a few small instances of the problem. What is usually more
difficult is to prove that a greedy algorithm yields an optimal solution (when
it does). One of the common ways to do this is illustrated by the proof given
in Section 9.1: using mathematical induction, we show that a partially
constructed solution obtained by the greedy algorithm on each iteration can be
extended to an optimal solution to the problem.

The second way to prove
optimality of a greedy algorithm is to show that on each step it does at least
as well as any other algorithm could in advancing toward the problem’s goal.
Consider, as an example, the following problem: find the minimum number of
moves needed for a chess knight to go from one corner of a 100 × 100 board to the diagonally opposite corner.
(The knight’s moves are L-shaped jumps: two squares horizontally or vertically
followed by one square in the perpendicular direction.) A greedy solution is
clear here: jump as close to the goal as possible on each move. Thus, if its
start and finish squares are (1,1) and (100, 100), respectively, a sequence of
66 moves such as

solves the problem. (The
number ** k** of two-move advances can be
obtained from the equation 1 + 3

The third way is simply to
show that the final result obtained by a greedy algorithm is optimal based on
the algorithm’s output rather than the way it op-erates. As an example,
consider the problem of placing the maximum number of chips on an 8 × 8 board so that no two chips are placed on the
same or adjacent— vertically, horizontally, or diagonally—squares. To follow
the prescription of the greedy strategy, we should place each new chip so as to
leave as many available squares as possible for next chips. For example,
starting with the upper left corner of the board, we will be able to place 16
chips as shown in Figure 9.1a. Why is this solution optimal? To see why,
partition the board into sixteen 4 × 4 squares as shown in Figure 9.1b. Obviously,
it is impossible to place more than one chip in each of these squares, which
implies that the total number of nonadjacent chips on the board cannot exceed
16.

As a final comment, we should
mention that a rather sophisticated theory has been developed behind the greedy
technique, which is based on the abstract combinatorial structure called
“matroid.” An interested reader can check such books as [Cor09] as well as a
variety of Internet resources on the subject.

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