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python slots functions Hashing In previous sections we were able to make improvements in our search algorithms by taking advantage of information about where items are stored in the collection with respect to one another.
For example, by knowing that a list was ordered, we could search in logarithmic time using a binary search.
This concept is referred to as hashing.
In check this out to do this, we will need to know even more about where the items might be when we go to look for them in the collection.
If every item is where it should be, then the search can use a single comparison to discover the presence of an item.
We will see, however, that this is typically not the case.
A hash table is a agree, lucky slot numbers not of items which are stored in such a way as to make it easy to find them later.
Each position of the hash table, often called a slot, can hold an item and is named by an integer value starting at 0.
For example, we will have a slot named 0, a slot named 1, a slot named 2, and so on.
Initially, the hash table contains no items so every slot is empty.
We can implement a hash table by using a list with each element initialized to the special Python value None.
In other words, there are m slots in the table, named 0 through 10.
Figure 4: Hash Table with 11 Empty Slots The mapping between an item and the slot where that item belongs in the hash table is called the hash function.
The hash function will take any item in the collection and return an integer in the range of slot names, between 0 and m-1.
Assume that we have the set of integer items 54, 26, 93, 17, 77, and 31.
Note that this remainder method modulo arithmetic will typically be present in some form in all hash functions, since the result must be in the range of slot names.
Table 4: Simple Hash Function Using Remainders Item Hash Value 54 10 26 4 93 5 17 6 77 0 31 9 Once the hash values have been computed, we can insert each item into the hash table at the designated position as shown in.
Note that 6 of the 11 slots are now occupied.
Figure 5: Hash Table with Six Items Now when we want to search for an item, we simply use the hash function to compute the slot name for the item and then check the hash table to see if it is present.
If everything is where it should be, we have found a constant time search algorithm.
You can probably already see that this technique is going to work only if each item maps to a unique location in the hash table.
Since 77 also had a hash value of 0, we would have a problem.
According to the hash function, two or more items would need to be in the same slot.
Clearly, collisions create a problem for the hashing technique.
We will discuss them in detail later.
Hash Functions Given a collection of items, a hash function that maps each item into a unique slot is referred to as a perfect hash function.
If we know the items and the collection will never change, then it is possible to construct a perfect hash function refer read article the exercises for more about perfect hash functions.
Unfortunately, given an arbitrary collection of items, there is no systematic way to construct a perfect hash function.
Luckily, python slots functions do not need the hash function to be perfect to still gain performance efficiency.
One way to always have a perfect hash function is to increase the size of the hash table so that each possible value in the item range can be accommodated.
This guarantees that each item will have a unique slot.
Although this is practical for small numbers of items, it is not feasible when the number of possible items is large.
For example, if the items were nine-digit Social Security numbers, this method would require almost one billion slots.
If we only want to store data for a class of 25 students, we will be wasting an enormous amount of memory.
Our goal is to create a hash function that minimizes the number of collisions, is easy to compute, and evenly distributes the items in the hash table.
There are a number of common ways to extend the simple remainder method.
We will consider a few of them here.
The folding method for constructing hash functions begins by dividing the item into equal-size python slots functions the last piece may not be of equal size.
These pieces are then added together to give the resulting hash value.
For example, if our item was the phone number 436-555-4601, we would take the digits and divide them into groups of 2 43,65,55,46,01.
If we assume our hash table has 11 slots, then we need to perform the extra step of dividing by 11 and keeping the remainder.
Some folding methods go one step further and reverse every other piece before the addition.
Another numerical technique for constructing a hash function is called the mid-square method.
We first square the item, and then extract some portion of the resulting digits.
You should verify that you understand how these values were computed.
Table 5: Comparison of Remainder and Mid-Square Methods Item Remainder Mid-Square 54 10 3 26 4 7 93 5 9 17 6 8 77 0 4 31 9 6 We can also create hash functions for character-based items such as strings.
To remedy this, we could use the position of the character as a weight.
The modification to the hash function is left as an exercise.
Figure 7: Hashing a String Using Ordinal Values with Weighting You may be able to think of a number of additional ways to python slots functions hash values for items in a collection.
The important thing to remember is that the hash function has to be efficient so that it does not become the dominant part of the storage and search process.
If the hash function is too complex, then it becomes more work compute the slot name than it would be to simply do a basic sequential or binary search as described earlier.
This would quickly defeat the purpose of hashing.
Collision Resolution We now return to the problem of collisions.
When two items hash to the same slot, we must have a systematic method for placing the second item in the hash table.
This process is called collision resolution.
As we stated earlier, if the hash function is perfect, collisions will never occur.
However, since this is often not possible, collision resolution becomes a very important part of hashing.
One method for resolving collisions looks into the hash table and tries to find another open slot to hold the item that caused the collision.
A simple way to do this is to start at the original click value position and then move in a sequential manner through the slots until we encounter the first slot that is empty.
Note that we may need to go back to the first slot circularly to cover the entire hash table.
This collision resolution process is referred to as open addressing in that it tries to find the next open slot or address in the hash table.
By systematically visiting each slot one at a time, we are performing an open addressing technique called linear probing.
When we attempt to place 44 into slot 0, a collision occurs.
Under linear probing, we look sequentially, slot by slot, until we find an open position.
In this case, we find slot 1.
Again, 55 should go in slot 0 but must be placed in slot 2 since it is the next open position.
The final value of 20 hashes to slot 9.
Since slot 9 is full, we begin to do linear probing.
We visit slots 10, 0, 1, and 2, and finally find an empty slot at position 3.
Figure 8: Collision Resolution with Linear Probing Once we have built a hash table using open addressing and linear probing, it is essential that we utilize the same methods to search for items.
Assume we want to look up the item 93.
When we compute the hash value, we get 5.
Looking in slot 5 reveals 93, and we can return True.
What if we are looking for 20?
Now the hash value is 9, and slot 9 is currently holding 31.
We cannot simply return False since we know that there could have been collisions.
We are now forced to do a sequential search, starting at position 10, looking until either we find the item 20 or we find an empty slot.
A disadvantage to linear probing is the tendency for clustering; items become clustered in the table.
This means that if many collisions gold slot videos at the same hash value, a number of surrounding slots will be filled by the linear probing resolution.
This will have an impact on other items that are being inserted, as we saw when we tried to add the item 20 above.
A cluster of values hashing to 0 had to be skipped to finally find an open position.
This cluster is shown in.
Figure 9: A Cluster of Items for Slot 0 One way to deal with clustering is to extend the linear probing technique so that instead of looking sequentially for the next open slot, we skip slots, thereby more evenly distributing the items that remarkable, sim slot tablet pc think caused collisions.
This will potentially reduce the clustering that occurs.
This means that once a collision occurs, we will look at every third slot until we find one that is empty.
Otherwise, part of the table will be unused.
To ensure this, it is often suggested that the table size be a prime number.
This is the reason we have been using 11 in our examples.
A variation of the linear probing idea is called quadratic probing.
In other words, quadratic probing uses a skip consisting of successive perfect squares.
Figure 11: Collision Resolution with Quadratic Probing An alternative method for handling the collision problem is to allow each slot to hold a reference to a collection or chain of items.
Chaining allows many items to exist at the same location in the hash table.
When collisions happen, the item is still placed in the proper slot of the hash table.
As more and more items hash to the same location, the difficulty of searching for the item in the collection increases.
Figure 12: Collision Resolution with Chaining When we want to search for an item, we use the hash function to generate the slot where it should reside.
Since each slot holds a collection, we read more a searching or alive demo dead slot to decide whether the item is present.
The advantage is that on the average there are likely to be many fewer items in each slot, so the search is perhaps more efficient.
We will look at the analysis for hashing at the end of this section.
Q-1: In a hash table of size 13 which index positions would the following two keys map to?
You need to use the hash table size as the modulo value.
Implementing the Map Abstract Data Type One of the most useful Python collections is the dictionary.
Recall that a dictionary is an associative data type where you can store key—data pairs.
The key is used to look up the associated data python slots functions />We often refer to this idea as a map.
The map abstract data type is defined as follows.
The structure is an unordered collection of associations between a key and a data value.
The keys in a map are all unique so that there is a one-to-one relationship between a key and a value.
The operations are given below.
It returns an empty map collection.
If the key is already in the map then replace the old value with the new value.
One of the great benefits of a dictionary is the fact that given a key, we can look up the associated data value very quickly.
In order to provide this fast look up capability, we need an implementation that supports an efficient search.
In we use two lists to create a HashTable class that implements the Map abstract data type.
One list, called slots, will hold the key items and a parallel list, called data, will hold the data values.
When we look up a key, the corresponding position in the data list will hold the associated data value.
We will treat the key list as a hash table using the ideas presented earlier.
Note that the initial size for the hash table has been chosen to be 11.
Although this is arbitrary, it is important that the size be a prime number so that the collision resolution algorithm can be as efficient as possible.
The put function see assumes that there will eventually be an empty slot unless the key is already present in the self.
It computes the original hash value and if that slot is not empty, iterates the rehash function until an empty slot occurs.
If a nonempty slot already contains the key, the old data value is replaced with the new data value.
Dealing with the situation where there are no empty slots left is an exercise.
If the value is not in the initial slot, rehash is used to locate the next possible position.
Notice that line 15 guarantees that the search will terminate by checking to make sure that we have not returned to the initial slot.
If that happens, we have exhausted all possible slots and the item must not be present.
The final methods of the HashTable class provide additional dictionary functionality.
This means that once a HashTable has been created, the familiar index operator will be available.
We leave the remaining methods as exercises.
First we will create a hash table and store some items with integer keys and string data values.
Note that the value for the key 20 is being replaced.
However, due to collisions, the number of comparisons is typically not so simple.
Even though a complete analysis of hashing is beyond the scope of this text, we can state some well-known results that approximate the number of comparisons necessary to search for an item.
This means that collision resolution is more difficult, requiring more comparisons to find an empty slot.
With chaining, increased collisions means an increased number of items on each chain.
As before, we will more info a result for both a successful and an unsuccessful search. python slots functions python slots functions python slots functions python slots functions python slots functions python slots functions

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