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of this sum gets computed in a xed amount of time The total running time for the addition algorithm is therefore of the form c 0 + c1 n, where c0 and c1 are some constants; in other words, it is linear Instead of worrying about the precise values of c 0 and c1 , we will focus on the big picture and denote the running time as O(n) Now that we have a working algorithm whose running time we know, our thoughts wander inevitably to the question of whether there is something even better Is there a faster algorithm (This is another persistent question) For addition, the answer is easy: in order to add two n-bit numbers we must at least read them and write down the answer, and even that requires n operations So the addition algorithm is optimal, up to multiplicative constants! Some readers may be confused at this point: Why O(n) operations Isn t binary addition something that computers today perform by just one instruction There are two answers First, it is certainly true that in a single instruction we can add integers whose size in bits is within the word length of today s computers 32 perhaps But, as will become apparent later in this chapter, it is often useful and necessary to handle numbers much larger than this, perhaps several thousand bits long Adding and multiplying such large numbers on real computers is very much like performing the operations bit by bit Second, when we want to understand algorithms, it makes sense to study even the basic algorithms that are encoded in the hardware of today s computers In doing so, we shall focus on the bit complexity of the algorithm, the number of elementary operations on individual bits because this accounting re ects the amount of hardware, transistors and wires, necessary for implementing the algorithm.

permanent These cells may also be affected as a result of atrophic rhinitis and local radiation therapy or by a very rare type of tumor (esthesioneuroblastoma) that originates in the olfactory epithelium There is also a group of rare diseases in which the primary receptor neurons are congenitally absent or hypoplastic and lack cilia One of these is the Kallman syndrome of congenital anosmia and hypogonadotropic hypogonadism A similar disorder occurs in the Turner syndrome and in albinos because of the absence of olfactory pigment or some other congenital structural defect Anosmia that follows head injury is most often due to tearing of the delicate laments of the receptor cells as they pass through the cribriform plate, especially if the injury is severe enough to cause fracture The damage may be unilateral or bilateral With closed head injury, anosmia is relatively infrequent (6 percent of Sumner s series of 584 cases) Some recovery of olfaction occurs in about one-third of cases over a period of several days to months Beyond 6 to 12 months, recovery is negligible Cranial surgery, subarachnoid hemorrhage, and chronic meningeal in ammation may have a similar effect Strangely, in some of the cases of traumatic anosmia, there is also a loss of taste (ageusia) Ferrier, who rst described traumatic ageusia in 1876, noted that there was always anosmia as well an observation subsequently corroborated by Sumner Often the ageusia clears within a few weeks A bilateral lesion near the frontal operculum and paralimbic region, where olfactory and gustatory receptive zones are in close proximity, would best explain this concurrence, but this has not been proven Obviously the interruption of olfactory laments alone would not explain ageusia In women, olfactory acuity varies throughout the menstrual cycle and may be disordered during pregnancy Nutritional and metabolic diseases such as thiamine de ciency, vitamin A de ciency, adrenal and perhaps thyroid insuf ciency, cirrhosis, and chronic renal failure may give rise to transient anosmia, all as a result of sensorineural dysfunction A large number of toxic agents the more common ones being organic solvents (benzene), metals, dusts, cocaine, corticosteroids, methotrexate, aminoglycoside antibiotics, tetracyclines, opiates, and L-dopa can damage the olfactory epithelium (Doty et al) It has been reported that a large proportion of patients with degenerative disease of the brain show anosmia or hyposmia, for reasons that are quite unclear Included in this group are Alzheimer, Parkinson, Huntington, and Pick disease and the Parkinsondementia syndrome of Guam The studies relating to this subject have been reviewed by Doty A number of theories have been proposed to explain these ndings, but they are conjectural It has been known for some time that alcoholics with Korsakoff psychosis have a defect in odor discrimination (Mair et al) In this disorder, anosmia is presumably due to degeneration of neurons in the higherorder olfactory systems involving the medial thalamic nuclei Hyman and colleagues have remarked on the early neuronal degeneration in the region of the hippocampus in cases of Alzheimer disease, but we know of no systematic studies of the central olfactory connections in this or any other degenerative disorder Anosmia has been found in some patients with temporal lobe epilepsy and particularly in such patients who had been subjected to anterior temporal lobectomy In these conditions, Andy and coworkers have found an impairment in discriminating the quality of odors and in matching odors with test objects seen or felt As with other sensory modalities, olfaction (and taste) are diminished with aging The receptor cell population is depleted, and if the loss is regional, neuroepithelium is slowly replaced with res-.

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