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It can be noted that the Boolean expression minimization is an NP-complete problem which an efficient exact algorithm is difficult to find. As shown in [7] and [8], Espresso-II is a fast and efficient heuristic algorithm used in practice. In [6], the authors give an efficient linear time solution to find the minimum Boolean expression for a range. An other well known technique for resolving the range matching problem specially in TCAM solutions is the range to ternary strings conversion. A ternary strings ft is of the form b;b2---b, with b; € {0,1,*}. Converting an arbitrary range into a minimum number of ternary strings is the Boolean expression minimization problem as shown in [6]. This technique gives 2w — 4 overlapping ternary strings. For better illustration of the conversion problem, we can use Karnaugh map technique (K-map) as a visual diagram to show all possibilities as shown in Figure 1. The K-map rule is to minimize the number of grouping and to maximize their sizes. Hence, the range [1, 14] can be expressed using only 4 ternary strings in 4 manners as follows: B. Non-Adjacent Form

Figure 1 It can be noted that the Boolean expression minimization is an NP-complete problem which an efficient exact algorithm is difficult to find. As shown in [7] and [8], Espresso-II is a fast and efficient heuristic algorithm used in practice. In [6], the authors give an efficient linear time solution to find the minimum Boolean expression for a range. An other well known technique for resolving the range matching problem specially in TCAM solutions is the range to ternary strings conversion. A ternary strings ft is of the form b;b2---b, with b; € {0,1,*}. Converting an arbitrary range into a minimum number of ternary strings is the Boolean expression minimization problem as shown in [6]. This technique gives 2w — 4 overlapping ternary strings. For better illustration of the conversion problem, we can use Karnaugh map technique (K-map) as a visual diagram to show all possibilities as shown in Figure 1. The K-map rule is to minimize the number of grouping and to maximize their sizes. Hence, the range [1, 14] can be expressed using only 4 ternary strings in 4 manners as follows: B. Non-Adjacent Form

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Figure 1. All possible K-map illustrations of the 4-bit range [1, 14]
Figure 1. All possible K-map illustrations of the 4-bit range [1, 14]
larger power of two range as follows: The proposed algorithm processes the ranges two by two and because of not all consecutive elementary ranges can be converted, we test if two consecutive ranges [/; , wi] and [/2 , ua] can be converted as follows: On one hand, if conversion is possible and the given ranges have same widths the resulting range is the union. On the other hand, we must check signs and the difference between widths to express two consecutive ranges as an exclusion of a power of two range from a larger power of two range.
larger power of two range as follows: The proposed algorithm processes the ranges two by two and because of not all consecutive elementary ranges can be converted, we test if two consecutive ranges [/; , wi] and [/2 , ua] can be converted as follows: On one hand, if conversion is possible and the given ranges have same widths the resulting range is the union. On the other hand, we must check signs and the difference between widths to express two consecutive ranges as an exclusion of a power of two range from a larger power of two range.
general case of the worst result is given by the following w-bit range as follows: Table II and Table III show some real world examples of port ranges and compare the number of prefixes given by the standard case (DRPC) with the number of prefixes given by the proposed NAF technique.
general case of the worst result is given by the following w-bit range as follows: Table II and Table III show some real world examples of port ranges and compare the number of prefixes given by the standard case (DRPC) with the number of prefixes given by the proposed NAF technique.
Figure 2. The NAF expansion technique vs the standard direct range-to-prefix conversion technique (DRPC): conversion of arbitrary w-bit ranges.
Figure 2. The NAF expansion technique vs the standard direct range-to-prefix conversion technique (DRPC): conversion of arbitrary w-bit ranges.
aeuewe ae NUMBER OF PREFIXES NEEDED FOR WELL KNOWN, REGISTERED AND DYNAMIC PORT NUMBER RANGES. Table III
aeuewe ae NUMBER OF PREFIXES NEEDED FOR WELL KNOWN, REGISTERED AND DYNAMIC PORT NUMBER RANGES. Table III
NUMBER OF PREFIXES NEEDED FOR IN/OUT PORT NUMBER RANGES OI SOME USED APPLICATIONS Table II
NUMBER OF PREFIXES NEEDED FOR IN/OUT PORT NUMBER RANGES OI SOME USED APPLICATIONS Table II
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