4.1 Symbol Tables. API sequential search binary search ordered operations. Symbol tables

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. ymbol Tables ymbol tables Key-value pair abstraction. Insert a value with specified key. Given a key, for the corresponding value. I sequential binary ordered operations x. D lookup.

www.yale.edu... www.harvard.edu...55 www.simpsons.com.5.5. lgorithms in Java, th dition obert edgewick and Kevin Wayne opyright October, :: key value ymbol table applications ymbol table I ssociative array abstraction.

1 . ymbol Tables ymbol tables Key-value pair abstraction. Insert a value with specified key. Given a key, for the corresponding value. I sequential binary ordered operations x. D lookup. Insert U with specified I address. Given U, find corresponding I address. U I address lgorithms in Java, th dition obert edgewick and Kevin Wayne opyright October, :: key value ymbol table applications ymbol table I ssociative array abstraction. ssociate one value with each key. application purpose of key value dictionary find definition word definition book index find relevant pages term list of page numbers file share find song to download name of song computer ID financial account process transactions account number transaction details web find relevant web pages keyword list of page names compiler find properties of variables variable name type and value routing table route Internet packets destination best route D find I address given U U I address reverse D find U given I address I address U genomics find markers D string known positions file system find file on disk filename location on disk public class T<Key, Value> T() void put(key key, Value val) Value get(key key) create a symbol table put key-value pair into the table (remove key from table if value is null) value paired with key (null if key is absent) void delete(key key) remove key (and its value) from table boolean contains(key key) is there a value paired with key? boolean ismpty() is the table empty? int size() number of key-value pairs in the table Iterable<Key> keys() all the keys in the table I for a generic basic symbol table a[key] = val; a[key]

onventions Keys and values Values are not null. ethod get() returns null if key not present. ethod put() overwrites old value with new value. Intended consequences. asy to implement contains().

Key type: several natural assumptions. ssume keys are omparable, use compareto(). ssume keys are any generic type, use equals() to test equality.

2 onventions Keys and values Values are not null. ethod get() returns null if key not present. ethod put() overwrites old value with new value. Intended consequences. asy to implement contains(). public boolean contains(key key) return get(key)!= null; an implement lazy version of delete(). public boolean delete(key key) put(key, null); Value type. ny generic type. Key type: several natural assumptions. ssume keys are omparable, use compareto(). ssume keys are any generic type, use equals() to test equality. ssume keys are any generic type, use equals() to test equality and hashode() to scramble key. Best practices. Use immutable types for symbol table keys. Immutable in Java: tring, Integer, Double, File, utable in Java: Date, tringbuilder, Url,... 5 T test client for traces T test client for analysis Build T by associating value i with ith string from standard input. public static void main(tring[] args) T<tring, Integer> st = new T<tring, Integer>(); tring[] a = tdin.readll().split("\\s+"); for (int i = ; i < a.length; i++) st.put(a[i], i); for (tring s : st.keys()) tdout.println(s + " " + st.get(s)); keys values 5 output 5 Frequency counter. ead a sequence of strings from standard input and print out one that occurs with highest frequency. % more tinytale.txt it was the best of times it was the worst of times it was the age of wisdom it was the age of foolishness it was the epoch of belief it was the epoch of incredulity it was the season of light it was the season of darkness it was the spring of hope it was the winter of despair % java Frequencyounter < tinytale.txt it % java Frequencyounter < tale.txt business % java Frequencyounter < leipzig.txt government tiny example ( words, distinct) real example (5,5 words,, distinct) real example (,,55 words, 5,5 distinct)

Frequency counter implementation public class Frequencyounter public static void main(tring[] args) int minlen = Integer.parseInt(args[]); T<tring, Integer> st = new T<tring, Integer>(); while (!tdin.

3 Frequency counter implementation public class Frequencyounter public static void main(tring[] args) int minlen = Integer.parseInt(args[]); T<tring, Integer> st = new T<tring, Integer>(); while (!tdin.ismpty()) ignore short strings tring word = tdin.readtring(); if (word.length() < minlen) continue; if (!st.contains(word)) st.put(word, ); else st.put(word, st.get(word) + ); tring max = ""; st.put(max, ); for (tring word : st.keys()) if (st.get(word) > st.get(max)) max = word; tdout.println(max + " " + st.get(max)); create T read string and update frequency print a string with max freq I sequential binary ordered operations equential in a linked list lementary T implementations: summary Data structure. aintain an (unordered) linked list of key-value pairs. T implementation earch. can through all keys until find a match. sequential (unordered list) Insert. can through all keys until find a match; if no match add to front. key value first red nodes are new gray nodes are untouched worst case average case hit ordered iteration? operations on keys / no equals() 5 black nodes are accessed in circled entries are changed values osts for java Frequencyounter < tale.txt using inkedistt hallenge. fficient implementations of both and. Trace of linked-list T implementation for standard indexing client

4 Binary Data structure. aintain an ordered array of key-value pairs. earch. Binary. I sequential binary ordered symbol table ops successful for lo hi m unsuccessful for Q keys[] lo hi m loop exits with lo > hi: return entries in black are a[lo..hi] entry in red is a[m] loop exits with keys[m] = : return Trace of binary for rank in an ordered array Binary : Java implementation Binary : mathematical analysis public Value get(key key) if (ismpty()) return null; int i = rank(key); if (i < && keys[i].compareto(key) == ) return vals[i]; else return null]; private int rank(key key) int lo =, hi = -; while (lo <= hi) int mid = lo + (hi - lo) / ; int cmp = key.compareto(keys[mid]); if (cmp < ) hi = mid - ; else if (cmp > ) lo = mid + ; else if (cmp == ) return mid; return lo; number of keys < key roposition. Binary uses ~ lg compares to any array of size. Def. T() number of compares to binary in a sorted array of size. T( / ) + left or right half Binary recurrence. T() T( / ) + for >, with T() =. ot quite right for odd. ame recurrence holds for many algorithms. olution. T() ~ lg. For simplicity, we'll prove when is a power of. True for all. [see O ] 5

Binary recurrence Binary : trace of standard indexing client Binary recurrence. T() T( / ) + for >, with T() =. roblem. To, need to shift all greater keys over. roposition.

.. stop applying, T() = T( / ) + + + + = lg + 5 keys[] 5 5 circled entries are changed values 5 5 5 5 5 5 5 5 entries in gray 5 did not move entries in red were ed vals[] 5 entries in black moved to

5 Binary recurrence Binary : trace of standard indexing client Binary recurrence. T() T( / ) + for >, with T() =. roblem. To, need to shift all greater keys over. roposition. If is a power of, then T() lg +. f. key value T() given T( / ) + T( / ) + + apply recurrence to first term T( / ) apply recurrence to first term... stop applying, T() = T( / ) = lg + 5 keys[] 5 5 circled entries are changed values entries in gray 5 did not move entries in red were ed vals[] 5 entries in black moved to the right Trace of ordered-array T implementation for standard indexing client lementary T implementations: summary worst case average case hit ordered iteration? operations on keys sequential (unordered list) / no equals() binary (ordered array) log log / yes compareto() T implementation 5 I sequential binary ordered operations osts for java Frequencyounter < tale.txt using OrderedrrayT hallenge. fficient implementations of both and.

Ordered symbol table I Ordered symbol table I min() get(::) floor(:5:) select() keys(:5:, :5:) ceiling(::) max() size(:5:, :5:) is 5 rank(::5) is keys values :: hicago :: hoenix :: ouston ::5 hicago

6 Ordered symbol table I Ordered symbol table I min() get(::) floor(:5:) select() keys(:5:, :5:) ceiling(::) max() size(:5:, :5:) is 5 rank(::5) is keys values :: hicago :: hoenix :: ouston ::5 hicago :: ouston :: hicago :: eattle ::5 eattle ::5 hoenix :: hicago :: hicago ::5 hicago :: eattle ::5 eattle :5:5 hicago :5: hicago :: eattle :: hoenix xamples of ordered symbol-table operations public class T<Key extends omparable<key>, Value> T() create an ordered symbol table void put(key key, Value val) put key-value pair into the table (remove key from table if value is null) Value get(key key) value paired with key (null if key is absent) void delete(key key) remove key (and its value) from table boolean contains(key key) is there a value paired with key? boolean ismpty() is the table empty? int size() number of key-value pairs Key min() smallest key Key max() largest key Key floor(key key) largest key less than or equal to key Key ceiling(key key) smallest key greater than or equal to key int rank(key key) number of keys less than key Key select(int k) key of rank k void deletein() delete smallest key void deleteax() delete largest key int size(key lo, Key hi) number of keys in [lo..hi] Iterable<Key> keys(key lo, Key hi) keys in [lo..hi], in sorted order Iterable<Key> keys() all keys in the table, in sorted order I for a generic ordered symbol table Binary : ordered symbol table operations summary sequential binary lg min / max floor / ceiling lg rank lg select ordered iteration log worst-case running time of ordered symbol table operations

ELEMENTARY SEARCH ALGORITHMS

ELEMENTARY SEARCH ALGORITHMS BB 22 - GOIT TODY DT. OF OUT GIIG ymbol Tables I lementary implementations Ordered operations TY GOIT cknowledgement: The course slides are adapted from the slides prepared by. edgewick and K. Wayne of