Really Efficient Use of Keys: the Trie Adding Item to a Trie Scrunching Example much about cost of comparisons. adding bat and faceplate. (unrelated to Tries on preceding slides) worst case is length of string. ticked. arrays, each indexed 0..9 should throw extra factor of key length, L, into costs: 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 a f A2: comparisons really means Θ(ML) operations. b ✷ trout pike ghee milk oil look for key X, keep looking at same chars of X M times. a bat f ✷ x a 3 4 5 6 7 8 9 better? Can we get search cost to be O(L)? b ab ax fa a b e o b c cumin mace multi-way decision tree, with one decision per character ✷ ✷ ✷ ✷ aba abbas axe axolotl fabric fac t them, but keep track of original index of each item: e ✷ A3: 0 1* 2 3 4 5* 6 7 8 9* abate face A2: 0 1 2* 3 4 5 6* 7* 8 9 h p t A1: 0* 1 2 3 4 5* 6 7* 8 9 0 -1 1 -1 2 5 5 7 6 7 9 ✷ faceplate✷ ✷ abash facet A123: bass trout pike ghee milk oil salt cumin mace 19:39:39 2018 CS61B: Lecture #31 2 19:39:39 2018 CS61B: Lecture #31 4 19:39:39 2018 CS61B: Lecture #31 6 CS61B Lecture #31 The Trie: Example A Side-Trip: Scrunching obvious implementation for internal nodes is array in- balanced search structures (DS(IJ), Chapter 9 abase, abash, abate, abbas, axolotl, axe, fabric, facet} character. L show paths followed for “abash” and “fabric” performance, length of search key. Pseudo-random Numbers (DS(IJ), Chapter 11) N internal node corresponds to a possible prefix. independent of , number of keys. Is there a depen- in path to node = that prefix. arrays are sparsely populated by non-null values—waste of a f a f arrays on top of each other! ✷ x b a empty) entries of one array to hold non-null elements of ab ax fa a b e o b c ✷ ✷ ✷ ✷ ✷ aba abbas axe axolotl fabric facet markers to tell which entries belong to which array. s t ✷ abas abate h abash✷ 19:39:39 2018 CS61B: Lecture #31 1 19:39:39 2018 CS61B: Lecture #31 3 19:39:39 2018 CS61B: Lecture #31 5 Probabilistic Balancing: Skip Lists Probabilistic Balancing: Skip Lists Probabilistic Balancing: Skip Lists can be thought of as a kind of n-ary search tree in which can be thought of as a kind of n-ary search tree in which can be thought of as a kind of n-ary search tree in which to put the keys at “random” heights. to put the keys at “random” heights. to put the keys at “random” heights. thought of as an ordered list in which one can skip large thought of as an ordered list in which one can skip large thought of as an ordered list in which one can skip large example: example: example: ⇓ ⇓ 25 30 40 50 55 60 90 95 100 115 120 125 130 140 150 ∞ 20 25 30 40 50 55 60 90 95 100 115 120 125 130 140 150 ∞ 20 25 30 40 50 55 60 90 95 100 115 120 125 130 140 150 ∞ start at top layer on left, search until next step would then go down one layer and repeat. start at top layer on left, search until next step would start at top layer on left, search until next step would above, we search for 125 and 127. Gray nodes are looked at; then go down one layer and repeat. then go down one layer and repeat. nodes are overshoots. above, we search for 125 and 127. Gray nodes are looked at; above, we search for 125 and 127. Gray nodes are looked at; the nodes were chosen randomly so that there are about nodes are overshoots. nodes are overshoots. nodes that are >k high as there are that are k high. the nodes were chosen randomly so that there are about the nodes were chosen randomly so that there are about >k k >k k searches fast with high probability. nodes that are high as there are that are high. nodes that are high as there are that are high. searches fast with high probability. searches fast with high probability. 19:39:39 2018 CS61B: Lecture #31 8 19:39:39 2018 CS61B: Lecture #31 10 19:39:39 2018 CS61B: Lecture #31 12 Practicum Probabilistic Balancing: Skip Lists Probabilistic Balancing: Skip Lists scrunching idea is cute, but can be thought of as a kind of n-ary search tree in which can be thought of as a kind of n-ary search tree in which to put the keys at “random” heights. to put the keys at “random” heights. good if we want to expand our trie. complicated. thought of as an ordered list in which one can skip large thought of as an ordered list in which one can skip large more useful for representing large, sparse, fixed tables many rows and columns. example: example: Furthermore, number of children in trie tends to drop drastically ⇓ gets a few levels down from the root. practice, might as well use linked lists to represent set of ∞ ∞ children.. 20 25 30 40 50 55 60 90 95 100 115 120 125 130 140 150 20 25 30 40 50 55 60 90 95 100 115 120 125 130 140 150 arrays for the first few levels, which are likely to have start at top layer on left, search until next step would start at top layer on left, search until next step would children. then go down one layer and repeat. then go down one layer and repeat. above, we search for 125 and 127. Gray nodes are looked at; above, we search for 125 and 127. Gray nodes are looked at; nodes are overshoots. nodes are overshoots. the nodes were chosen randomly so that there are about the nodes were chosen randomly so that there are about nodes that are >k high as there are that are k high. nodes that are >k high as there are that are k high. searches fast with high probability. searches fast with high probability. 19:39:39 2018 CS61B: Lecture #31 7 19:39:39 2018 CS61B: Lecture #31 9 19:39:39 2018 CS61B: Lecture #31 11 Probabilistic Balancing: Skip Lists Probabilistic Balancing: Skip Lists Probabilistic Balancing: Skip Lists can be thought of as a kind of n-ary search tree in which can be thought of as a kind of n-ary search tree in which can be thought of as a kind of n-ary search tree in which to put the keys at “random” heights. to put the keys at “random” heights. to put the keys at “random” heights. thought of as an ordered list in which one can skip large thought of as an ordered list in which one can skip large thought of as an ordered list in which one can skip large example: example: example: ⇓ ⇓ ⇓ 20 25 30 40 50 55 60 90 95 100 115 120 125 130 140 150 ∞ 20 25 30 40 50 55 60 90 95 100 115 120 125 130 140 150 ∞ 20 25 30 40 50 55 60 90 95 100 115 120 125 130 140 150 ∞ start at top layer on left, search until next step would start at top layer on left, search until next step would start at top layer on left, search until next step would then go down one layer and repeat. then go down one layer and repeat. then go down one layer and repeat. above, we search for 125 and 127. Gray nodes are looked at; above, we search for 125 and 127. Gray nodes are looked at; above, we search for 125 and 127. Gray nodes are looked at; nodes are overshoots. nodes are overshoots. nodes are overshoots. the nodes were chosen randomly so that there are about the nodes were chosen randomly so that there are about the nodes were chosen randomly so that there are about nodes that are >k high as there are that are k high. nodes that are >k high as there are that are k high. nodes that are >k high as there are that are k high. searches fast with high probability. searches fast with high probability. searches fast with high probability. 19:39:39 2018 CS61B: Lecture #31 14 19:39:39 2018 CS61B: Lecture #31 16 19:39:39 2018 CS61B: Lecture #31 18 Probabilistic Balancing: Skip Lists Probabilistic Balancing: Skip Lists Probabilistic Balancing: Skip Lists can be thought of as a kind of n-ary search tree in which can be thought of as a kind of n-ary search tree in which can be thought of as a kind of n-ary search tree in which to put the keys at “random” heights. to put the keys at “random” heights. to put the keys at “random” heights. thought of as an ordered list in which one can skip large thought of as an ordered list in which one can skip large thought of as an ordered list in which one can skip large example: example: example: ⇓ ⇓ ⇓ 20 25 30 40 50 55 60 90 95 100 115 120 125 130 140 150 ∞ 20 25 30 40 50 55 60 90 95 100 115 120 125 130 140 150 ∞ 20 25 30 40 50 55 60 90 95 100 115 120 125 130 140 150 ∞ start at top layer on left, search until next step would start at top layer on left, search until next step would start at top layer on left, search until next step would then go down one layer and repeat.